Geometric Constraints on Human Speech Sound Inventories

We investigate the idea that the languages of the world have developed coherent sound systems in which having one sound increases or decreases the chances of having certain other sounds, depending on shared properties of those sounds. We investigate the geometries of sound systems that are defined by the inherent properties of sounds. We document three typological tendencies in sound system geometries: economy, a tendency for the differences between sounds in a system to be definable on a relatively small number of independent dimensions; local symmetry, a tendency for sound systems to have relatively large numbers of pairs of sounds that differ only on one dimension; and global symmetry, a tendency for sound systems to be relatively balanced. The finding of economy corroborates previous results; the two symmetry properties have not been previously documented. We also investigate the relation between the typology of inventory geometries and the typology of individual sounds, showing that the frequency distribution with which individual sounds occur across languages works in favor of both local and global symmetry.

Furthermore, as noted in Study 2, the uniform random feature inventories, which differ only in their size distribution and not in their composition across the four classes, do not show substantial differences in central tendency for Econ. The practical significance of the sensitivity of the scale to inventory size is mitigated in these sets, compared to what the impact would be if Econ were equally likely to take on any of its logically possible values for a given size. We see this in Figure 2, which compares the median and mean Econ in uniform random feature inventories from Study 2 (all pooled together) to those in sample of Econ values, matching the distribution of sizes, in which some number of features is simply drawn uniformly at random from the range of possible values for a given size, and Econ is computed from that. Intuitively, these are hypothetical inventories that are equally likely to fall on any of the lines in Figure 1. Natural inventories are given for reference. There is always instability for small inventory sizes, after which the random Econ values see their mean fall off as a function of size, and their median sharply so. For random inventories, however, this pattern is attenuated; Econ falls off a bit, but remains relatively stable. These inventories actually tend to have relatively large levels of Econ, and thus the dramatic shrinking of the scale for the smaller values of Econ does not always pose a problem. For the natural inventories, this is more of an issue, and, across the board, the limitation of full economy to powers of two will clearly limit our ability to compare inventories of different sizes.

IMPACT OF USING CONTRASTIVE FEATURES ON LOC AND GLOB
All three measures assume that that only the irreducible dimensions ("contrastive features") are relevant to the geometry. In the case of Loc, adding in non-contrastive features would change the matrix of distances (hence, the geometry). For natural segments there is a fair amount of redundancy in the feature representation we use, so this would substantially reduce the number of pairs with distance equal to one. A measure of something similar could be constructed by replacing all the distances between pairs of sounds with their rank, and defining oppositions to be pairs with minimum distance (whatever that is for the given inventory). In the case of Glob, variant representations are more relevant. The calculation of Glob when non-contrastive features are added would remain unchanged, except that additional terms would be added to the sum.

RELATION TO THE CONTRASTIVE HIERARCHY
Many phonological theories not only claim that non-contrastive features have little or no impact on lexical storage in a language, but also make the further claim that there is a cognitively impactful order on features (a "contrastive hierarchy"). A contrastive hierarchy per se does not play a direct role in any of our statistics, but all three can be interpreted through the lens of a contrastive hierarchy. On a binary feature representation, a contrastive hierarchy for an inventory induces a maximum-binary branching tree where each leaf is a single sound. The pair of nodes under the root split are the sounds marked [-] and [+] respectively for the first feature in the hierarchy. If a node contains only one sound, it has no children; if it contains more than one sound which all have the same value for the next feature in the hierarchy then it has one child node, equal to the same set of sounds; otherwise, it has two children, respectively, the sounds bearing [-] and [+] for the next feature in the hierarchy.
From this perspective, Econ is constructed using the median of all possible tree depths, given all possible sets of contrastive features for the inventory. Each set of contrastive features yields only one tree depth, regardless of the contrastive hierarchy over those features. Loc and Glob, on the other hand, can be seen as combining information from different contrastive hierarchies for a given set of contrastive features.
Loc considers each contrastive feature in turn and imagines it at the bottom of a contrastive hierarchy. It measures how many branching nodes there would be at this bottom level of the tree. If the inventory were perfectly economical, then there would be 2 p−1 branching nodes, where p is the number of features. It then sums all these numbers and computes a normalized rank over the possible values among trees of a given depth with a given number of leaf nodes, a step which can be seen as a way of expressing indifference to or uncertainty about the ordering of features in the hierarchy. The final step takes the median over all analyses into contrastive features, which can be seen as a way of expressing indifference to or uncertainty about the set of contrastive features.
Glob considers each contrastive feature in turn and imagines it at the top of a contrastive hierarchy. It measures the imbalance between the number of sounds in each of the two nodes at the top level. It then sums all these numbers and computes a normalized rank over the possible values among trees of a given depth with a given number of leaf nodes, with a given number of branching nodes (which restricts the shape of the tree further in such a way as to take into account the fact that the number of branching nodes is informative about how much imbalance there could be). This step can again be seen as a way of expressing indifference to or uncertainty about the ordering of features in the hierarchy. The final step takes the median over all analyses into contrastive features, which again can be seen as a way of expressing indifference to or uncertainty about the set of contrastive features.
To actually exploit the idea of a contrastive hierarchy and incorporate it into inventory geometry, a measure would need to be constructed that considers only a single order at once and combines (different) information from the different levels of the hierarchy.

VARIANT REPRESENTATIONS
The number of variant representations is small for some inventories, and large for others, with a large spread (between one and 14,640). Generally speaking, the number of variant representations for larger inventories is larger, and the number can be very large for random feature inventories.
To assess the risk of taking the median-that is, the risk that this might yield unreasonably high values among the variant representations-we also performed a replication of Study 1 and Study 2 in which we drew a variant representation uniformly at random rather than taking the median. The results of this are exactly the same, qualitatively, as in the main text, except that the difference in Glob for whole inventories is no longer evident. The results are summarized in Table 1. Table 1. Mean values of Econ, Loc, and Glob, with bootstrap AUC 95% intervals for comparisons of distributions in a replication of Study 1 and Study 2 in which a variant representation was drawn uniformly at random. The first rows are the means. AUC intervals above 0.5 indicate that the inventory group on the right side of the < has systematically larger values for the given geometry statistic, and AUC intervals below 0.5 (in italics) indicate that the inventory group on the left side of the < actually has systematically larger values for the given geometry statistic. Cases where the interval includes 0.5 are in parentheses.           ũː -+ + + + --+ --+ + + + -+ -+ ---+ + ṳː -+ + + + --+ + -  anterior  labial  back  high  ATR  lateral  distributed   LONG  spread  low  syllabic  continuant   nasal  sonorant  coronal  tense  constr  vocalic  strident  EXTRA  consonantal   voice  round bɮ  tˠ, ǃ, ǃd, ǃq, ǃr, ǃs, ǃt, ǃx, ǃz, ǃʃ, ǃʒ m|G, n|, |, |n, |n̥ ɡǁ, ɡǁkx, ɡǁx