Matrix diagrams

Provenience

The arrangement of things in a table--a matrix arrangement--appears to be as old as written records; lists or inventories arranged in tables figure prominently among the Linear В tablets, for example. In linguistics, the matrix is perhaps as old as Indian grammatical theory. The characters of the devanāgarī alphabet⁵ are usually given in a matrix arrangement. The arrangement of the symbols for stops and nasals given by Whitney as

(Whitney 1889:2)

and by Allen as

(Allen 1953:20)

expresses a classification of the sounds based on their analysis according to articulatory features. Neither Whitney nor Allen says whether or not the matrix arrangement occurs in the ancient works; but the sequence (given by the numbers beside the characters in Whitney's figure) of the letters is that of the varna-samāmnāya. Cardona points out (personal communication) that, in any case, "we cannot literally speak of a written chart: such lists were always recited only."

We are on firmer ground, then, with the analysis presented by the figure than with the figure itself. Sounds with the same place of articulation are a varga, which Allen translates "class," and Whitney, "series." Such subgroups were useful for generalizations: Bloomfield (1929:271) cites Pānini's practice of referring to a group of letters by the first letter of the group and the "silent letter" following it in the samāmnāya sequence. Sounds with the same manner of articulation are indicated by numbering alike corresponding members of the five classes (Allen, 47). Thus the two dimensions of the matrix are defined just as they are for modern articulatory phonetics. For instance, Whitney's first column, which is the same as Allen's first row, is the set of voiceless unaspirated stops / k c t ṭ p / --a set of sounds sharing the same manner'of articulation; in a modern articulatory matrix, they would constitute a single row, /p t ṭ к/. Whitney's first row, which is the same as'Allen's first column, is the set of velars /к k^h g gh ŋ/--a set of sounds sharing the same place of articulation; in a modern articulatory matrix, they would constitute a single column. In the matrix derived from the varna-samämnäya, then, rows and columns are sets of sound alike in place or manner, sets of sounds sharing particular features: for the set /k c ṭ t p/, these features are [-Voice],[ -Tense], and [-Continuant]. The varna-samāmnāya of the Hindu grammarians is in fact a componential analysis. In Prague School terms, the set of twenty-five consonants may be factored into series (Whitney's column; Allen's row) and order (Whitney's row; Allen's column). The set of voiceless stops /k c ṭ t p/ is a series; the set of velars /k k^h g g^h ŋ/ is an order.

Until the second half of the nineteenth century, matrix diagrams are rare. Exceptions are in works so little known that their authors are among those dubbed by Abercrombie (1948) ״forgotten phoneticians.״ Francis Lodwick, in his "Essay Towards an Universal Alphabet'' of 1648, is one of these:

(Abercrombie 1948:9)

Here as elsewhere the matrix implies a componential analysis.

In the late nineteenth and early twentieth centuries, with the interest in analphabetic notation--which, by definition, carries with it a componential analysis--that appears to have begun with Alexander Melville Bell's Visible Speech, the matrix gained wider currency. It serves in Bell's work as the ground on which are displayed the analphabetic characters:

(Bell 1867:110)

This Bell labels (1867:110) ״table of English elements, showing their position in the universal alphabet.״ The matrix arrangement interprets the notation: it is taken for granted that placing the symbols on the ground provided by the matrix-- showing their positions in a "universal," or at any rate a fuller, inventory of sounds--somehow translates a far from accessible notation. Henry Sweet employs the matrix in a like manner for his " 'Organic' (revised Visible Speech) notation" (1890:vii):

(Sweet 1890:77)

The reader's impulse to leaf through the book until he comes upon just such a presentation argues that it renders the system intelligible-- in outline if not in detail--that it is a map the reader can carry with him into the analphabetic wilderness.⁶

How can a figure seldom met explain symbols even less familiar? We can resolve this by backtracking a little to examine the concept of phonological space. Moulton (1962:24-25) traces it back as far as Grimm's Deutsche Grammatik, and finds it implicit in subsequent works like Hermann Paul's Prinzipien der Sprachgeschichte (1880) and Karl Luick's Untersuchungen zur Lautgeschichte(1896) . The notion of phonological space in the linguistics of the nineteenth century accounts for the appearance, in the works of Bell and Sweet, of the matrix. Phrases like those Moulton finds in Grimm (" 'occupy a position,' " " ' move from a position,'" "' fill a gap' ") and in Paul ("' directions' in which phonemes 'move'"), metaphoric or not, presuppose a matrix arrangement of sounds. This matrix arrangement must have underlain the notion of phonological space, if not on paper then at least in the minds of its proponents.

Meaning

The matrix expresses a single meaning, componential analysis. It does not, like the tree, comprise several meanings; but it does comprise two graphic variants whose difference lies in how the data are displayed on the figure. We shall leave them for the moment, to consider the matrix as a design.

The design of the matrix rests on three graphic elements: rows, columns, and cells. All have their meaning by virtue of contrast. Contrast, indeed, is what makes the figure. By virtue of the contrast between the two dimensions, rows and columns have their meaning: a row or column incorporates the meaning of the dimension along which it lies and the meaning of the dimension on which it is a locus. Thus, in Whitney's figure, the first row means Mall and only the velar consonants": "all" because it lies along the horizontal dimension, "only" because it is a locus on the vertical. Similarly, the first column means "all and only the voiceless unaspirated consonants": "all" because it lies along the vertical dimension, "only" because it is a locus on the horizontal. A cell is simultaneously a locus on both dimensions; its meaning is thus the product of the meanings ascribed to the two dimensions. It is because of this that Bell's figure can translate his notation. Bell defines each symbol as its position in the universal alphabet--the universal alphabet being the matrix arrangement of all speech sounds, and position being meant literally, as the location of a particular sound in this arrangement. We know by its location that p, for example, is to be translated [p]--because it is plotted on the row labelled (by means of the key word key) "voiceless," and the column labelled "lip":

Conversely, the meanings assigned to the two dimensions may be inferred if the value of the sound in any cell is known. Labels are then unnecessary: this is how we are able to read Lodwick's diagram of the "universal alphabet." And because every cell is the intersection of the two dimensions, it is possible to determine what would be in an empty cell, were it filled; indeed, not only possible, but inescapable. The empty cells in this diagram from Trubetzkoy, for instance,

(Trubetzkoy 1939:153)

are clearly /p^h Ø ŋ ŋ/ .

The matrix figure, as we have seen, has no top or bottom, left or right. Its meaning is the same, whether a given matrix is

The matrix, unlike the tree, has no directionality. It is for this reason that the meaning of empty cells can be inferred--or rather, both the lack of directionality and the possibility of inferring the meaning of empty cells are the result of a characteristic of the design which we shall take up later. For the time being, note that in a matrix diagram, as on a map, every point in space is a priori defined, and labels need not be included. The latitudes and longitudes, so to speak, are given; and place names, though handy, are not necessary. We may think of matrix diagrams as maps of phonological space.

Componential analysis in a matrix

Though the meaning of matrix diagrams is always componential analysis, there are two ways in which they are used to express this meaning. The first we shall call the matrix; the second, the grid. All of the examples we have looked at so far are of the first sort. For its use in modern articulatory phonetics, the matrix has a key something like this:

(Pike 1947:195 [phonetic symbols omitted])

However, this sort of diagram appears for articulatory phonetics as early as Sievers's work (see following page). The notion of componential analysis is implicit in such a matrix, just as the technique of componential analysis is implicit in articulatory phonetics.

Componential analysis is explicit in the use of matrix diagrams in the work of Trubetzkoy. Nearly all of the figures in the Grundzüge are matrix diagrams. They show not only the inventory of phonemes but also their pattern--the system of

(Sievers 1881:106)

(Pike 1974:232)

correlations in which they stand. Empty cells, as well, give information about pattern. The absence of /p^hø ŋ ŋ./ in Trubetzkoy1s diagram, for instance, signifies the absence of the contrasts [±Continuant] and [±Nasal] in part of the system--a statement about pattern. Here the boundary of the figure is also exploited. The position of /r/ "ausserhalb des Korrelationsystems" (Trübetzkoy 1939 : 153) is represented by putting it precisely there. Outside the boundary is outside the pattern.

There are, of course, differences among matrix diagrams. First, particular matrices differ in domain. The area mapped by one matrix may be larger or smaller or altogether different from that mapped by another. A diagram like Pike's (on page 48) for the sounds of a particular language does not map the same area as the matrix arrangement for the International Phonetic Alphabet (at right), the domain of which is all speech sounds. Secondly, particular matrices differ in ostensible purpose. The matrix of Bell and Sweet is a glossary; that of the IPA, an inventory; that of Trubetzkoy, an explicit representation of system or pattern. In the first sort, the sounds are ostensibly of no account; in the second sort, the sounds are simply themselves; in the third sort, the sounds are more than themselves, for a matrix representation of a system of correlations breaks down the sounds into their components and at the same time builds out of them the structure of a sound system.

Finally, particular matrices differ in design: modification of the matrix figure may follow upon the purpose for which it is use.d. For instance, the configuration can be tailored to the pattern of the sound system represented--as in another figure from Trubetzkoy (Cairns 1972:922):

(Trubetzkoy 1939:64)

This diagram permutes both columns (the usual order of which follows the oral cavity) and rows. It is thereby a better fit for the data, the system of correlations:

(Sound above the heavy line are fricatives sounds in shaded areas are voiced.)

None of these differences, however, alters the meaning of matrix diagrams. The juxtaposition of the two dimensions, the cell as a locus at the intersection of the two dimensions, carry the same meaning. But the second type of matrix diagram, which we have called the grid, is different; though it, like the matrix diagrams we have looked at, expresses componential analysis, it construes the graphic elements of the matrix in a different way.

Componential analysis in a grid

An example of a grid is Halle's representation for the sounds of Russian--a translation, in fact, of the key reproduced in the last chapter:

(Halle 1959:45)

A grid has the graphic elements of a matrix but construes them differently. Rows, in the grid as in the matrix, represent features; columns, however, represent not features but sets of feature specifications, and a cell contains not a unit of the sound system, but a partial specification of a unit. The intersection of the two dimensions no longer defines a whole entity, but simply characterizes one entity, a sound, with respect to another, a feature:

The stuff of the matrix--the units of the sound system--has been moved out to the periphery.

Why this rearrangement? It allows representation of combinations of more than two features at a time. If the units of a sound system fill the cells, a unit is the product of two and only two features, one on each dimension. (Often, it is true, manner of articulation combines more than one feature--say, [±Voice] and [±Continuant]: these we shall consider portmanteau features and treat as single features.) If, on the other hand, the units of a sound system are deployed along the periphery, there is theoretically no limit to the number of features specifying a unit. This gain is offset by a greater loss. In a grid, an empty cell signifies no more than the fact that part of the specification of a sound is dispensable; and the boundary of a grid cannot be invested with meaning. Because nothing can be placed outside the figure, because the figure has no end, the di- chotomy between inside and outside vanishes. The contrasts on which the matrix is founded--the juxtaposing of rows and columns, of full and empty cells, of inside and outside--are not exploited in the grid. And at the same time that it fails to exploit these graphic elements, the grid requires additional graphic devices. Labelling is essential: one cannot take soundings by the values filling the cells to find out the meaning of the dimensions.

The net effect of these changes in the meaning of the graphic elements is that the figure no longer says the same thing. Shifting the units of the sound system from center to periphery means the figure no longer makes its statement about them. What fills the cells is the topic of the figure: in the grid it is not the sounds, not the features, but the values of the features.

Both the inventory and the pattern of sounds are thus relegated to a lesser place. The grid in fact gains the capacity to specify a theoretically infinite number of features and to define a theoretically infinite number of sounds, at the expense of its capacity to represent pattern. A grid is less a graphic representation than a list. It is simply a list that happens to be in two dimensions, stretching across the page as well as down.

Expanding the chart of figures and meanings to accommodate matrix diagrams (recalling that matrix and grid are simply different expressions of the same meaning) gives us

Now besides the homonymy exhibited by tree diagrams--one figure expressing four meanings--we have synonymy as well; for matrix diagrams and tree diagrams both express componential analysis. The correspondence between form and meaning for graphic representation in linguistics become even more tangled when we consider still another kind of diagram.