Groucho Marxism

In linguistics, the term ‘phoneme’ refers to any of the perceptually distinct units of sound in a specified language that distinguish one word from another; for example p, b, d, and t in the English words pad, pat, bad, and bat. Languages vary considerably in the number of phonemes they have, from as few as 9 in the Brazilian indigenous language Pirahã to as many as 141 in the southern African language ǃXũ. It is usually claimed that there are 44 phonemes in English: 24 consonant phonemes and 20 vowel phonemes. However the Hungarian linguist Peter Szigetvári has recently argued – convincingly, in my view – that English has just 6 vowel phonemes, and that the remaining 18 vowel sounds can be considered combinations of these six vowels plus a glide (y, w, or h).

This suggests that the number of vowel phonemes may have been overestimated in other languages too. A recent (2013) study by the Chinese linguist San Duanmu seems to bear this out. Duanmu argues – again, convincingly in my view – that vowel inventories in all the world’s languages can be represented using just four basic features, which we may take as [low], [front], [round], and [raised] (here I am following the standard convention of putting features within square brackets []). What makes Duanmu’s argument convincing is that he has tested his hypothesis against data on languages from around the world, using two separate data sources (the databases UPSID and P-Base). Duanmu’s analysis puts an upper bound of 16 on the number of vowel phonemes a language could possible have.

This means that any vowel in any of the world’s languages can be represented by a 4-vector (a,b,c,d), where a, b, c, and d represent the features [low], [front], [round], and [raised], and can be either 0 or 1. A 1 signifies that the feature is present in that vowel phoneme, and a 0 signifies it is absent. We can then define four basic vowels as A = (1,0,0,0), I = (0,1,0,0), U = (0,0,1,0), and G = (0,0,0,1). Any vowel phoneme in any of the world’s languages can then be represented using combinations of these four basic vowels. For example, we can define the compound vowels E = A+I = (1,1,0,0), O = A+U = (1,0,1,0), and Y = I+U = (0,1,1,0). Under this conception, the space of all vowels is represented by a mathematical structure called a tesseract, or 4-dimensional hypercube.

In the linguistics literature, the vowel space is usually represented as a quadrilateral, based on the shape of the tongue when pronouncing different vowel sounds. However the British linguist Geoff Lindsey and others have argued that the vowel space is better represented as a triangle, based on the resonant frequencies of different vowel sounds. Thus, the quadrilateral representation is based on the articulation of different vowel sounds, whereas the triangular representation is based on their acoustic characteristics. The representation of the vowel space as a tesseract in theory provides an alternative view. Unfortunately, it is impossible for us 3-dimensional creatures to visualize a 4-dimensional shape such as a tesseract.

What we can do is transform a 4-dimensional shape into 3 dimensions, and again into 2 dimensions if we like, both of which we can visualize. One way of doing this is by taking what is known as the ‘vertex figure’ of the 4-dimensional shape. Roughly speaking, this is the figure exposed when a corner of a general polytope – that is, a figure with flat faces – is sliced off. The vertex figure of a tesseract is a tetrahedron, the 3-dimensional analogue of a triangle. The orthographic projection of a tetrahedron into 2-dimensional space results in a quadrilateral in general, or a triangle when viewed from a face or vertex. Thus, the representation of the vowel space as a tesseract provides a way to reconcile the two different 2-dimensional representations found in the literature.

The representation of the vowel space as a tesseract also provides a way to formalise a phonological theory known as Element Theory. The basic idea here is that all phonemes are made up of combinations of elements or phonological primes, which in the context of vowels are usually taken as A, I, and U, plus one other element which we are calling G. Element Theory has a number of versions and has since its inception in the mid-1980s been reformed in various ways with the aim of reducing the element inventory, to avoid over generation (being able to generate more structures than attested cross-linguistically). The empirical work of San Duanmu has now demonstrated that only four elements are required to represent the vowel phonemes of all the world’s languages.

It is remarkable that Element Theory apparently existed for 30 years before anyone bothered to check the data to determine how many elements were actually needed. And that was just for vowel phonemes; as far as I know, the number of elements required to represent the consonant phonemes of all the world’s languages is still an open question. I will return to this question in a future blog post.