This volume consists of eighteen essays based on papers given at a conference with the same title as the publication held at the Thomas-Institut, University of Cologne, Germany, in the summer of 1997. Five essays are written in English, one in French and twelve in German. The publication addresses fundamental questions concerning the concept of musica in the Middle Ages and its position in the mathematical quadrivium and concerning the understandings of how mathematics and the musical discipline were connected and separated and, finally, also about the changes in music philosophy during the long period that we still denote with the problematic name of the Middle Ages.
Considering the origin of the book, it is not very surprising to find a volume which contains many interesting and even important contributions and still as a volume does not seem quite satisfactory since its many contributions form a rather scattered whole. Maybe this is also to some extent due to the organization of the volume. The book begins with a section on "sound or number" (three papers by Cecilia Panti, Eva Hirtler and Frank Hentschel), then follows a section on " concordia, proportio, consonantia" (containing two papers by Stephen Gersh and Michael Walter), a section on musica in commentaries and glosses (five papers by Andreas Speer, Udo R. Jeck, Christian Meyer, Alison M. Peden, and Calvin M. Bower), two papers labelled under the common heading of "the pedagogical context" (Matthias Hochadel and Max Haas), a section titled "the use of philosophical ideas in a practice based context" (papers by Wolfgang Hirschmann and Klaus-Jurgen Sachs), a section on musica and metaphysica (papers by Klaus Wolfgang Niemoller and Jan A. Aertsen) and finally "interdisciplinary patterns of thought" (Andre Goddu and Lawrence Gushee). The books further contains a bibliography and an index.
Even though this organization in itself seems logical enough (and consciously chosen as Frank Hentschel points out in his Foreword), the papers themselves do not corroborate this since some papers easily could have fitted in to several of the categories, and since the chronological order of the presentation, because of the chosen order, is upset in a way that at least to me seems curious. To me it would have seemed a better introduction to the fundamental questions raised in many of the articles to have started with the papers taking up the medieval reception of the music philosophy of the Greek antiquity and the early Middle Ages. Not the least Calvin M. Bowers' interesting discussion of (mainly) the 9th-century reception of Boethius under Louis the Pious--based on the earliest preserved manuscripts of and glosses to the De institutione musica--gives an insight into the distinctions and connections between the mathematical and musical disciplines which it would have been fruitful to have read before the opening sections.
These mainly deal with the question of how the concept of musica gradually takes in also less abstract, concrete physical understandings of sound (in the modern sense), in other words how the treatment of musica to some (but not always the same) extent is loosened from the Boethian strict mathematical foundation. In many ways it seems that the materials and ideas laid out in this book could have led to a-- much needed--detailed, historical treatment of the medieval philosophy of music. This is not meant as a criticism of the present publication, obviously such a project--even though many of the building blocks seem to be present already in this volume--is of quite another scope than the publication of a series of conference papers. It is much more meant as an encouragement to carry out such a project in the interdisciplinary way that seems intended, and commented upon by Frank Hentschel when he makes the very basic, and important statement (in the Foreword) that the modern "Schemata der Wissenschaften" do not coincide with those of the Middle Ages (p. xi). But what to me seems mostly needed in this area would be a grand synthesis of the (changing) medieval concept of musica, treated--as here fragmentarily--in relation to the quadrivium, to the arithmetics of the time(s) in particular, and also--in spite of the lack of reference between these areas--to the musical practice of the time.
It is almost a commonplace--and confirmed here by Calvin M. Bower concerning the 9th-century music philosophy (p. 182)-- that the early medieval music practise had basically no influence on the philosophical concept of music. This could lead to strange propositions as the one quoted by Prof. Bower from a 9th-century gloss to Boethius that the proportion 9:8-- the step of a (whole) tone --is a common measure for all intervals, a proposition obviously false in a practical context where the semitone could not be left out of the argument as easily as in a theoretical one. The problem of the semitone is more directly addressed by Alison M. Peden in the previous-- somewhat loosely organized--article (clearly not very well polished as she essentially repeats a remark made on p. 152 about the structuring of the contents of Macrobius^Ò 4th-century Commentary on Cicero's Somnium Scipionis on p. 159). She points to the consequences of the Boethian (and Pythagorean) understanding of the intervals (including the claim that the tone cannot be divided in two equal intervals) creating problems for the definition of the semitone (especially pp. 153 and 158). Her discussion thus overlaps somewhat with the discussion of this particular problem by Frank Hentschel who points out new ways of approaching the concept of musica in the 14th century where Johannes Boen (writing in 1357) involves a much more empirical approach to sound in his discussion of the question about the possibility of an equal division of the (whole) tone, thus arriving at a different conclusion than Boethius (especially pp. 39-42 and 59-60).
Before this--in the very first essay of the volume--Eva Hirtler has made the more general point of the transition of the concept of musica from a purely (Pythagorean and Boethian) mathematical abstract concept to a somewhat more ambiguous concept involving, in various degrees, a more material Aristotelian understanding, musica secundum auditum (p. 26). She discusses treatises from the 13th century (from the Boethian statements of Robert Grosseteste through, among others, Robert Kilwardby to the most advanced departure from the Boethian tradition in remarks by Albertus Magnus, and the somewhat more conservative views--in this respect--of Thomas Aquinas). Compare to this also Jan A. Aertsen's discussion of the Speculum musicae (Jacobus Leodiensis or Jacobus von Luttich) from the early 14th century which--as also Klaus Wolfgang Niemoller's treatment of 9th- century John Scot Eriugena--deals with more general metaphysical questions. Here the universalization of the concept of music in the Speculum musicae is discussed both in terms of its Aristotelian usage, its Boethian reception, and its theologically informed use of the concept of laus divina to describe yet another aspect of music.
The many reflections about the mathematical (and arithmetic) nature of the understanding of musica in the book might sometimes have profited from a more detached (anachronistic) mathematical description of what is going on. For instance, both in Frank Hentschel's and other discussions of the mentioned Pythagorean-Boethian postulate concerning the impossibility of an equal division of the tone, it did not surface very clearly for at least my mind what the heart of the problem actually was, until Hentschel finally gives the clue (p. 43) that the "true semitone" does not exist as it cannot be found in the proportions, "it belongs"--translating a part of Hentschel's statement--"to the conditions of a consonant that it is possible to represent it in a proportion of numbers". In other words, the problem is that the existence of the "true semitone" (about which modern man has no doubts) as an acceptable part of the system is dependent on its possible expression as a rational number. But that is not possible as a rather simple calculation can show. So the problem mathematically is simply the question of the extension from the set of integers and rational numbers to what later times describe as "real numbers", the continuum which was known and understood also in the Middle Ages, but only as a geometrical fact, at the time not possible to express in arithmetics. Considering Frank Hentschel's own (above quoted) ambition concerning interdisciplinarity, such perspectives might have been involved to outline the history of music philosophy in its relation to the history of mathematics to a higher degree.
Considering the rather involved arguments in the individual contributions it is not possible to comment on each of these. I cannot conclude my somewhat scattered remarks about individual contributions to this volume, however, without pointing to at least three more, quite interesting articles, very different mutually as well as from those mentioned hitherto: Max Haas' highly original and thought-provoking critical inquiry about the pedagogical-rhetorical background for the musica enchiriadis (9th century), Andre Goddu's linking medieval music philosophy to the emergence of modern science with its heliocentric world view of Copernicus and Kepler, and Lawrence Gushee's biographical discussion of Jehan des Murs.
In spite of the differences in style, approach, and overall purpose of the individual contributions, and the unavoidable differences also in terms of the interest and perspective of these--but such assessments in themselves are necessarily subjective and certainly more than anything reflect the interests and academic stance of the reviewer--I do want to underline that every single paper in the volume is of interest. The volume altogether, in spite of my above mentioned hesitations caused by its unevenness and lack of integration, certainly constitutes a welcome contribution to the literature on the medieval understanding of music.