The aim of Historical Glottometry is to provide a representation of language genealogy in which

- subgroups are defined by exclusively-shared innovations (following Leskien’s principle [1876], one of the mainstays of the Comparative Method);
- subgroups are allowed to intersect (as expected under the Wave model);
- the “strength” of each subgroup is measured on a continuous scale (rather than subgroups simply being absent or present).

How might one go about creating such a representation? One option would be to use an *isogloss map*, as in dialectology, but with the following properties:

- Each isogloss delimits a set of languages defined by one or more exclusively-shared innovations — that is, a
*genealogical subgroup*(in the sense of the Comparative Method); - The graphic representation of each isogloss can be used to represent the strength of the subgroup. For example, the contour’s thickness could be made proportional to the number of
*exclusively-shared innovations*.

However, this would fail to capture the intuition that the strength of a subgroup is not just defined by the number of innovations that *confirm* it, but is also affected by the number of innovations that *contradict* it (that is, the number of isoglosses that cross-cut the subgroup). If there are no innovations that cross-cut a subgroup, then its strength should be equal to the number of exclusively-shared innovations defining it; but as the number of cross-cutting innovations grows, the strength of the subgroup should be reduced accordingly, and ultimately tend to zero.

# Cohesiveness

The definition of **subgroupiness** that we use captures this intuition by weighting the number of exclusively-shared innovations (called ε ‘epsilon’) by a “**cohesiveness coëfficient**” (κ ‘kappa’). Consider a candidate subgroup. If the number of cross-cutting innovations is equal to zero, this means we have a “perfect” subgroup which is never contradicted: in this case, the coefficient κ is equal to 1. Conversely, as the number of cross-cutting innovations tends to infinity, κ should tend to zero.

Given a candidate subgroup S, let us call ** p** the number of innovations that properly confirm its cohesion (i.e., those which contain the whole of the subgroup as well as at least one language outside the subgroup); and

**the number of cross-cutting innovations. Together,**

*q**p*and

*q*constitute all the relevant evidence needed to assess a subgroup’s cohesiveness. We define κ as follows:

p + 1 κ = ————————— p + q + 1

Note that κ is always between 0 and 1, and can be expressed as a percentage.

The intuition behind this definition of cohesiveness is as follows. Consider a perfectly tree-like language family of the form [* a* [

*]]. In such a family, if we find an innovation that*

**b c***b*shares with

*a*, we can be absolutely certain that it is also shared with

*c*—otherwise the innovation would be incompatible with the tree. Indeed, this is the essential property of a tree: that if a member of a subgroup shares an innovation with a non-member, it

*always*“invites” all of the other members. Cohesiveness simply measures the proportion of the time that subgroup members are “loyal” in inviting all of their co-members (

**), relative to the number of times that they are “disloyal” in excluding some of their co-members (**

*p***).**

*q*# Subgroupiness

Given this definition of cohesiveness, we define the **subgroupiness** of a subgroup (called ς ‘sigma’). In principle, subgroupiness could simply be conceived as the product of *epsilon* and *kappa* — that is, the number of exclusively shared innovations, weighted by the subgroup’s cohesiveness rating:

( p + 1 ) ς = ε · κ = ε · (——————————) (p + q + 1)

However, we wish to add a final refinement to the results, by adding a **strictness parameter***n*, that would set the “penalty” for overlapping subgroups:

n n ( p + 1 ) ς = ε · κ = ε · (——————————) (p + q + 1)

The strictness parameter *n* ranges from 0 to ∞. As n → ∞, κ^{n} → 0 for all 0 ≤ κ < 1—which means that the only subgroups that are “left standing” are those that are perfectly cohesive, i.e., those that are never cross-cut, and thus conform to the “ideal” of a branch in a tree. Conversely, as n → 0, κ^{n} → 1 for all 0 < κ ≤ 1; in other words, ε·κ^{n} → ε, meaning that subgroupiness is measured simply by counting exclusively-shared innovations, no matter how often the subgroup may be cross-cut. By default, we set n = 1, though this is purely a matter of convenience, and carries no theoretical significance.

# Glottometric diagrams

Having measured the subgroupiness of every subgroup that is defined by at least one exclusively-shared innovation, we retain only those subgroups for which ς > 1. The reason for this is that since 0 < κ^{n} ≤ 1, the only condition under which ς > 1 is when ε > 1, i.e., when the subgroup is defined by more than one exclusively-shared innovation. Thus, this criterion excludes “hapaxes” (subgroups that are defined by just one innovation), as well as subgroups that are “no better than a hapax”: those which, despite being defined by more than one exclusively-shared innovation, have too low a cohesiveness value. “Better-than-a-hapax” subgroups may be seen as *well-supported genealogical subgroups*, knowing that the quality of that support can be made more demanding by adjusting the strictness parameter *n*.

Finally, well-supported subgroups can be represented on a schematic isogloss map in which line thickness is proportional to subgroupiness (ς); this is what we call a **glottometric diagram**. The isoglosses in a glottometric diagram may form one or more “connected sets”; we say that each connected set of isoglosses delimits a linkage.

Here is an example of a glottometric diagram: it represents the genealogy of the 18 northernmost languages of Vanuatu (see also our *Case studies* page), after the breakup of their shared ancestor Proto Oceanic.