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# Space

Further indexing power is obtained if we add ``space'' to ``time''. If only one event or element could occur at a time, as in considering any single time series, such as a waveform, the character output of a typewriter, the location of an automobile through the course of a day, etc., then mere linear ordering (for ``normal time'' this requires some fixed unit of measurement) of the events would be enough to orchestrate them completely. But if events may occur simultaneously, as when multiple time series are considered abstractly, another property of time becomes crucial: Each time serves as an index linking all the events (or states, or elements) occurring at that time. A many-to-one mapping between events and times is what is necessary for this more powerful indexing system, which we may call ``multi-linearity''.

It is a further, perhaps unnecessary, stipulation that there are a fixed number of ``places'' each of which must index some element at each and every time. Some representations may require an unstructured simultaneity of elements, rather than a fixed, ordered list of required elements. (This difference is analogous to the difference between variable-record-length signal representations and fixed-record-length signal representations).

In phonology, if the organization of sequences of phonological feature matrices is of the former, un-structured sort, then the features present at a given time (in addition to their values) may change. On the other hand, if such matrix-sequences are of the latter sort, then we may speak of fixed ``places'' or feature-locations, each of which, at every time, indexes some event, state or element -- or in the phonological universe, some feature-value. In the case of a fixed set of time-series, we may speak of place-structured simultaneity; on the other hand if we have a different, arbitrary listing of elements for each time, we have unstructured simultaneity. One may imagine various degrees and forms of partly-structured simultaneity, as for example if there is a regular alternation between sets of elements. Phonological constituents such as syllables, feet, phrases, etc., may be thought of as describing the structure of the alternation of feature inventories across time. Autosegmental tiers are simply locations in phonological space. In the rhythmic alternation across syllables or other phonological constituents, the inventory of features varies over constituent, so that, say, consonant features occur in onsets, vocalic ones in codas, etc. If features, or tiers, are simply locations in phonological space, then the alternation across syllables from one feature-inventory to another is a sort of rhythmic appearance and disappearance of the locations.

This shows limits to the space-and-time metaphor. As the vocal tract moves in speech production, different places become relevant to sound production at different times within syllable structure. Appearing and disappearing might rather be thought of as coming-into-relevance and falling-out-of-relevance. The different articulators that constitute the different ``locations'' in phonological space become relevant and irrelevant to sound production in association with syllable structure and other rhythmic constituents. This reflects the structure of physical actions in speech production as much as it reflects phonological structure. The patterns of rhythmic spatio-temporal structure in phonology is derived directly from a schematic view of the physical vocal tract. The tongue body features associated with vowels are not relevant to bilabial stop production, so those features can be considered nonexistent (or inactive) at the point of specifying the stop.

In any case, the task of orchestrating the events of multiple simultaneous time series requires a way of relating the events of one series to those of another: this implicitly requires what I here call finite-time-and-place indexing machinery, or multilinearity. Thus, to take a special case again, all events occurring now are linked in time; they ``temporally overlap'', or ``occur at the same time'', or ``are indexed by the same time index'', and also all events occurring in just one of the time series ``occur at the same place'', ``spatially overlap'', or ``are indexed by the same place index''. To summarize the universality property of multi-linear indexing in one predicate-calculus statement:

C: Each element which occurs is located at some place and at some time.

Next: Linguistic Time Up: Features of Time and Previous: Time
Thomas Veatch 2005-01-25