For the research on the topic of medieval notation and more specifically square notation, some of the more interesting textual sources appear to be late medieval theoretical writings on music. One of the activities of Nota Quadrata is to sift through the enormous corpus of such texts and to take note of any passages related, either directly or indirectly, to square notation. Thanks to the wonders of modern technology this task is proving to be less daunting than it might have been a few decades ago. Most of the medieval texts on music theory are now accessible in electronic format, thanks to an online database called Thesaurus Musicarum Latinarum, a growing collection of all known Latin music treatises developed by the Center for the History of Music Theory and Literature at Indiana University.

Our plan is to make select passages from the theoretical writings available on our website, with the original Latin texts parallel with English translations, supplemented by source information, comments, and so on. It is worth noting that only very few of these texts on notation have ever been translated into English. Thus one of the tasks is to come up with translations - something that will hopefully facilitate the work of scholars in the field. Syntactically the Latin of most medieval musical theorists is not very complex. The main difficulty in these treatises usually arises with terminology. Not only are some terms obscure, but also the medieval writers are not always consistent with words for notation. Usage may differ from century to century, place to place, author to author. From the thirteenth century onwards we encounter certain new terms of notation, some borrowed from other languages (Arabic, Greek, vernacular) that prove to be so unique that they appear only once or twice in the entire corpus of Latin musical treatises. All these issues highlight the need to keep track of various usages of music notation terms. As no comprehensive source on Latin music notation terminology is currently available, we have started compiling a word-list of Latin terms related to musical notation. Besides these the list will also contain any Latin words related to the work of a notator, such as writing tools (parchment, ink). For each entry are provided English definitions and examples of usage from medieval theoretical works on music (with abbreviated references in parentheses). On the page "Theorist biographies" are presented short descriptions of the most important medieval treatises on music theory from the thirteenth to the fifteenth centuries that deal with the writing of music. Information on each treatise includes biographical data about the authors - assuming we are fortunate enough to identify the author.

In order to get a better idea what sort of information concerning notation these theoretical writings may yield, let us consider one work as an example. Speculum musicae (The Mirror of Music) by Jacques de Liège is the largest surviving medieval treatise on music, a true summa of the theory and practice of music as understood in the later Middle Ages. It contains 521 chapters arranged in seven books. The work was probably written between 1330 and 1340, a peaceful short period before the Black Death struck and the Hundred Years War started to affect much of Western Europe. The mission of Jacques, as he states in the introduction, was to expose the errors of the Ars Nova composers (that is, Philippe de Vitry, Johannes de Muris and their circle) and to demonstrate the validity of the traditional teachings of the Ars Antiqua or older music (Franco of Cologne, Magister Lambertus and some others). Jacques de Liège has been often seen by historians as a reactionary figure. Others argue that he was a man solidly linked to the academic tradition of the time, one who emphasized the link between music and mathematics, and who was convinced that the musical tradition can only be developed on a traditional theological-philosophical foundation. The largest part of Jacques' encyclopedic work is devoted to musica speculativa, a realm of highly theoretical issues, which would nowadays more often be put in the category of philosophy and history of science, rather than musicology. The text draws upon Aristotle, Boethius, Isidore of Seville and others. This type of material may at first glance seem irrelevant to notation, but we should not assume that no link exists, let's say, between the mathematical theory of proportionality and various aspects of notation.

Now it is in the sixth book of the Speculum that Jacques turns to the issues of practical music such as modes and psalm tones. And only a small portion of this book deals directly with our main interest - mainly chapters 72 and 73. Jacques's treatment of music notation reflects a keen awareness of its development from the Greek origins up to the author's day. The important stages of Western notation such as Boethius' system of letter notation and the Guidonian method are described and lauded as progressive (historical accuracy of his specific claims, of course, is another matter). Jacques says that notation develops just as any art or craft moves from imperfect towards perfect in imitation of nature (ars de imperfecto tendat ad perfectum ad imitationem naturae). Here he is obviously not following the Darwinian concept of natural selection, but rather the Aristotelian movement of a substance towards its final cause ( e.g., just like an acorn becomes a full-grown oak-tree, so does the art of musical notation develop towards its own perfection). Each new stage, including the introduction of square notation, is described by words like perfectior, clarior, facilior (more perfect, clearer, simpler) - terms reflecting the scholastic ideals of the period found also in contemporary treatises on other subjects such as theology. The older or less desirable ways of musical notation are described by Jacques as imperfectior, confusior, incertior, more imperfect, more confusing, more unclear. These terms - both positive and negative - give us some idea about the main criteria by which the late medieval cantors and notators appraised the various methods of notation; musici cantores and notatores are the two types of people identified by Jacques as those who must know the craft. The author of our treatise appears to be in touch with the recent developments in notation both in France and elsewhere. For example, we learn that, as the French are leading the way with square notation, the Germans are still employing the old Guidonian system of colours, especially in secular (that is, non-monastic) churches. Jacques also describes the properties of square notation and distinguishes them from those of non-square notation. Moreover, he distinguishes the type of square notation that is used in plainchant from the one used in mensural music, and considers it wrong to mix these two up - implying, of course, that this was sometimes practiced.

Certain passages in the Speculum musicae by Jacques de Liège deal with the symbolic meaning of a square shape (quadratum) and the number four (numerus quaternarius) - two words etymologically related in Latin. It is well known that in medieval iconography certain shapes are often associated with symbolic meaning, for example, a triangle with Trinity or an octagon, the traditional shape of a baptismal font, with eternity. Then what about the square shape, a strikingly prominent feature of the new style of notation? Is it just a practical feature of a simpler and clearer style of notation or does it possibly carry some more profound symbolic meaning obvious to medieval musicians, but not to us? In his treatise Jacques is certainly sensitive to the characteristics of various shapes. In a context not directly related to notation he claims that just as a round object is movable and unstable, a square object is stable, solid, and as if immovable. Solidity is indeed characteristic not only to notation but also other aspects of the thirteenth-century culture. For example, the use of a broad pen in the script heightened the impression of weight and solidity, echoing to some extent the Gothic architecture of the period. Moreover, the Speculum teems with references to numerology and symbolism. In the same book, an entire chapter is dedicated to the importance and various meanings of number four. Number four is here associated with the four elements (fire, water, air, earth), the four cardinal virtues (prudence, justice, temperance, courage), the four humours of mankind (phlegmatic, choleric, melancholic, sanguine). In short, we learn that the whole harmony of the world and of music is based on this number. Is it then possible that square notation as a medium of music reflects this fourfold nature of harmony that Jacques is talking about? Although no circumstantial evidence in the text of Speculum directly connects this data on numerology to the shape of the square note, it is more than likely that a certain amount of association was habitual.