The basic theory of measurement based on a concatenation operation defined on pairs of elements of a continuous ordered set is presented when this operation can be viewed as halving of intervals, intervals being defined as ordered pairs of elements. Starting out with a series of assumptions, a sequence of conclusions is obtained which culminates in the existence and uniqueness theorems concerning a numerical representation. An alternative set of basic assumptions contains an autodistributivity requirement instead of a bisymmetry postulate. The results were obtained in a strict but rather undemanding way. Some remarks concerning scale types and related issues are appended. It appears doubtful whether a definition of scale types by means of a certain group of admissible transformations captures the psychological significance of a given scale well enough.
