Mensuration and Proportion Signs: Origins and Evolution.

The author describes her useful and provocative book as a report on what music theorists during the period ca. 1320 to ca. 1560 have to say about mensuration and proportion signs (loosely speaking: signs that tell the singer how the composer has organized rhythm). The book opens up as many avenues for further research as it sums up from past research, and for this reason it will be a useful reference work. It is written with authority and the presentation is very clear - not an easy accomplishment for a book devoted so rigorously to one of the most daunting topics in European music history.

But the book is much more than a report on what music theorists have to say. Each of the main chapters features a central problem and a creative hypothesis for solving it. The book is overflowing with fresh insights, recastings of old debates, new ways to extrapolate evidence, realignments of theoretical traditions, improved translations, etc. Readers of this journal will be pleased to discover a good bit of interdisciplinary material, especially in Chapter 2, which argues for the importance of Roman fractions (and systems of measuring, generally) for the early history and eventual evolution of how these signs were conceived, and also in Chapter 6, which argues for the importance of commercial arithmetic in theoretical reforms. The interdisciplinary dimension makes a very fresh contribution to the topic.

Obviously, technical details surrounding proportions affect performance of the polyphonic music that we know and love. To note only one of many contributions: the notion of diminutio per tertiam partem will no longer be a possibility, thanks to a stunning interpretation of sixteenth-century German theorists as having misread Johannes de Muris, ca. 1340. Technical details also shape our thinking about a piece and its place in history. For example, "O2," used as a modus cum tempore sign, dates from music that is probably from the 1440s and may be by Du Fay. (Earlier use of this sign meant something quite different.) If Du Fay invented this usage, then this may strengthen our growing sense of a relationship between him and the younger Busnoys, who used the sign often and distinctively but was unlikely (though there is no strong evidence against this) composing music in the 1440s.

Part of the fun in working with fifteenth-century music is the need, in forging coherent interpretations, to juggle evidence that is only vaguely located. Many interpretations in this book are boldly creative and they are always clearly presented - undoubtedly, they will inspire further debate. To take a central example: Tinctotis has been considered by some as the main representative of the venerable Franco-Flemish tradition, read - as the pre-eminent authority on mensural practices that feature concepts such as minim equivalence. The author considers Tinctoris not in this way but as a reformer (he is even labeled "Southern" at one point), one inspired not by the venerable premises of ars nova but by commercial arithmetic; he is viewed as out of step with the mainstream, which had come to place breve equivalence at the center of the proportional system through a synthesis of French and Italian practice. Whatever comes of this hypothesis, fifteenth-century studies will benefit from its exploration, presented here in an important book on music and culture.

Thomas Brothers DUKE UNIVERSITY