For a given family, corpus, style, or genre of musical rhythms, the pulse saliency histogram counts the relative frequency with which an onset occurs in each pulse position of the rhythm timespan (cycle, measure). Thus for a given fixed timespan a collection of rhythms may be expressed in its pulse saliency histogram (the empirical estimate of its probability distribution). The similarity of one group of rhythms to another group of rhythms or to theoretical models such as GTTM and its modifications may be measured by the degree to which their corresponding pulse saliency histograms are associated. The GTTM profile is characterized with four well-defined mathematical properties: alternations of strong and weak beats, pulse weight distribution, sub-symmetries, and fractal hierarchy. These properties provide quantitative methods for assessing the degree and nature of the structural similarities of rhythm corpora between each other and with regard to theoretical models of meter. Their usefulness is illustrated in the context of exploring the similarity of African and Western musical rhythms in terms of the quantity and type of meter they possess. In addition, several properties of metric hierarchies are uncovered that may characterize some of the differences between Renaissance music, Common Practice music, and German folk songs, as well as shed light on the complex evolution of meter in Western music and the relation of music to language.