The Distribution of Phoneme Frequencies across the World's Languages: Macroscopic and Microscopic Information-Theoretic Models

Fermín Moscoso del Prado Martín; Suchir Salhan

cs.CLMar 3, 2026

The Distribution of Phoneme Frequencies across the World's Languages: Macroscopic and Microscopic Information-Theoretic Models

Authors: Fermín Moscoso del Prado Martín, Suchir Salhan

Organizations: Department of Computer Science and Technology, University of Cambridge, UK

Abstract

We demonstrate that the frequency distribution of phonemes across languages can be explained at both macroscopic and microscopic levels. Macroscopically, phoneme rank-frequency distributions closely follow the order statistics of a symmetric Dirichlet distribution whose single concentration parameter scales systematically with phonemic inventory size, revealing a robust compensation effect whereby larger inventories exhibit lower relative entropy. Microscopically, a Maximum Entropy model incorporating constraints from articulatory, phonotactic, and lexical structure accurately predicts language-specific phoneme probabilities. Together, these findings provide a unified information-theoretic account of phoneme frequency structure.

Figures & tables

Figure 1: Rank-frequency plots for the five phoneme frequency distributions discussed in Sigurd (1968) , compared with the word frequency distribution in the Brown corpus. Note the logarithmic scales.

Figure 2: Geographic and genetic diversity of the languages included in the UDHR dataset.

Panel of Figure 3 — Figure 3: (a) Rank-frequency plot for the American English phoneme frequency distribution (red dots and lines) discussed in Sigurd (1968) , overlaid with the predicted rank-frequency distribution for symmetric Dirichlet distributions with estimated concentration parameters $\hat{\alpha}$ . The green lines are the predicted mean order statistics, the darker green shading denotes their standard deviations, and the lighter green shading denotes their 95% C.I. (b) Relationship between the size of the phonemic inventory (horizontal axis; note the logarithmic scale) with the estimated value of the concentration parameter $\hat{\alpha}$ (vertical axis) across the three datasets. Each point denotes the phonemic inventory for a language variety. The black line plots a doubly logarithmic linear regression (whose parameters are given in Equation 1 ) and the shaded area is its 95% C.I. (c) Rank-frequency plot for the American English phoneme frequency distribution (red dots and lines) discussed in Sigurd (1968) , overlaid with its rank-frequency distribution reconstructed from its inventory size using Equation 1 . The green lines are the predicted mean order statistics, the darker green shading denotes their standard deviations, and the lighter green shading denotes their 95% C.I.

Panel of Figure 4 — Figure 4: (a) Distributions (kernel density estimates) of the estimated values of the maximum entropy Lagrange multipliers across the UDHR datasets. (b) Comparison between the observed (red) and maximum entropy guessed (green) probabilities of the phonemes in the Abkhaz UDHR dataset (the horizontal axis plots the ranks of the observed data). (c) Comparison between the maximum entropy guessed (horizontal axis) and observed (vertical axis) probabilities of the phonemes across all UDHR datasets. The solid line plots a non-linear regression (lowess) and the dashed line plots the identity (note logarithmic scales). (d) Comparison between the maximum entropy guessed (horizontal axis) and observed (vertical axis) relative entropies of the phoneme distributions across all UDHR datasets. The solid line plots a linear regression (the shading denotes its 95% C.I.) and the dashed line plots the identity.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Abkhaz	Haitian Creole	Ossetian
Albanian	Hawaiian	Palauan
Arabela	Hiligaynon	Quechua (Cuzco)
Arabic	Hindi	Romanian
Asturian	Hmong	Samoan
Aymara	Hungarian	Sinhala
Basque	Iloko	Slovak

Appendix

Table 5

Linguistic Typology

Linguistic typology compares languages to identify patterns in their sounds, grammar, meaning, and history.

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Modelling the Diachronic Emergence of Phoneme Frequency Distributions

Fermín Moscoso del Prado Martín, Suchir Salhan

Department of Computer Science and Technology, University of Cambridge, UK

Linguistic Typology Heavy-Tailed Distributions

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Phoneme frequency distributions exhibit robust statistical regularities across languages, including exponential-tailed rank-frequency patterns and a negative relationship between phonemic inventory size and the relative entropy of the distribution. The origin of these patterns remains largely unexplained. In this paper, we investigate whether they can arise as consequences of the historical processes that shape phonological systems. We introduce a stochastic model of phonological change and simulate the diachronic evolution of phoneme inventories. A naïve version of the model reproduces the general shape of phoneme rank-frequency distributions but fails to capture other empirical properties. Extending the model with two additional assumptions -- an effect related to functional load and a stabilising tendency toward a preferred inventory size -- yields simulations that match both the observed distributions and the negative relationship between inventory size and relative entropy. These results suggest that some statistical regularities of phonological systems may arise as natural consequences of diachronic sound change rather than from explicit optimisation or compensatory mechanisms.

Abstract

Linguistic Typology

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