Many other treatments of vowel height take the opposing view, that height is a single phonological dimension. Such treatments can straightforwardly account for processes which increase or decrease vowel height by one step, as for example NYC English raising of tense low vowels /æh, oh/ (as in man and thought), or Jamaican Creole English raising of long mid vowels /e:, o:/ (FACE, GOAT) (cf. also processes referred to in Lindau 1978).
If vowel height is a single dimension, then there are many ways to distinguish the points on the scale, all of which amount to different approaches to a theory of numbers. Some (e.g., Ladefoged 1982:262-263; Labov, Yaeger, & Steiner 1972:167ff) use the traditional numbers, 1,2,3,4,..., of base n, where n is greater than the number of points on the scale.
Integers written in any base may be converted to base-2 numbers. Since generative phonologists prefer binary features, they have made a number of attempts to rewrite the familiar decimal scale in terms of various binary numbering systems. In all of these, various combinations of homogeneous binary features -- which amount to binary digits -- are used to construct a number theory for the vowel-height dimension.
In Clements' (1989) treatment of vowel height, binary features are used in an interesting way. There, the first binary feature distinguishes among low and non-low vowels; the second among the non-low vowels, etc. Thus the aperture (or height) dimension is organized by dividing the higher height into two. (i e æ) becomes ((i e) æ); (i e æ) becomes (((i e) ) æ). The second [open] feature cannot occur in a given language unless the first does. The second is added in order to divide the non-low vowels into two groups; the third is added to divide the higher of the non-low classes into two groups. This can be done in a language-particular way, so that if two heights pattern together, the nth [open] feature can be used to distinguish them; while the remaining n-1 [open] features classify them together.
Like other numbering systems, Clements' [open] features ``stack'', so that the nth [open] feature can only occur if the n-1th [open] feature occurs. But unlike Schane's, discussed below, they stack inwards, not at the end. This may be clearer if we convert the representations to binary numbers. Clements' system of aperture features may be represented as binary numbers with n binary digits (technically termed bits) representing the n [open] features. The set of resulting binary numbers are a restricted subset of the possible binary numbers that can be distinguished using n bits.
Considering 0 as [-open] and 1 as [+open], a set of 3 heights are represented as (0., 10, 11); 4 heights are (0..,10.,110,111), 5 heights are (0...,10..,110.,1110,1111). The periods represent bits which are filled in as 0's by a special redundancy rule. After the application of this redundancy rule to a 4-height set of vowels, the resulting binary numbers are those in Table .
Further redundancy remains to be factored out, since all digits can be predicted from a single feature specification: the location of the first 0, or, conversely, the location of the first 1. Whichever system is chosen, all other feature-values can be predicted from a single feature. In this system, a given language has a fixed number of these features, or bits; each segment is specified (on the surface) for all values of all the features.
How might a system of privative features work? If a feature were absent, i.e., if the bit were zero, then it would be absent. This is the case in the written representation of normal arithmetic numbers, where leading zeroes are not written.
In fact, a system with these properties has been proposed. In Schane's particle phonology (1984) the privative, additive, aperture feature, called a, may be thought of as a binary digit in just this way. The front vowels [i, e, , æ], are represented as i, ia, iaa, iaaa. Abstracting the height dimension and considering the aperture particles as binary digits, the numbers are as in Table .
Thus each added aperture particle adds another power of two to the number representing vowel height. Schane's particles are added to the end of the stack of aperture particles, but they could just as well be added at any place in the middle, thus accounting for the language-particular differences which Clements represents by inserting digits at various places.
All of these treatments of vowel height remain controversial. However, as shown here, they can all be seen as attempted theories of numbers. While this seems on the face of it to be rather remote from phonological theory, phonological feature theory can itself be seen as characterizing an inventory of named bits, with particular combinatory restrictions and phonetic interpretations, that together determine a set of possible numbers, which represent the phonological segments of language.
Next, I will instead present a novel formulation that is different in kind from the above systems, that also solves the impossible-combination problem of SPE's features [+high], [+low].