Cotto is rather indifferent and, of course, dry about the whole subject of organum. It occupied but a chapter in his rather long treatise. But the ‘Anonymus’ is full of enthusiasm and loud in his praises of this method of part-singing and bold in his declaration of its superiority over the unaccompanied plain-song. Such enthusiasm smacks a little of the layman, and is but another indication of the real origin of organum in the improvised descant of the people, quite out of the despotism of theory. The Anonymus gives a great many rules for the conduct of the organizing or improvising voice. He has divided the system into two modes, determined by the interval at which the voices start out. For instance, rules of the first mode state how the organizing voice must proceed when it starts in unison with the cantus firmus, or at the octave. If it starts at the fourth or fifth it is controlled by the rules of the second mode. There are three other modes which are determined by the various progressions of the parts in the middle of the piece. The division into modes and the rules are of little importance, for it is obvious that only the first few notes of a piece are definitely influenced by the position at which the parts start and that after this influence ceases to make itself felt the modes dissolve into each other. Thus, though the enthusiasm of the Anonymus points to the popularity of the current practice of organizing, whatever it may have been, his rules are but another example of the inability of theory to cope with it. Still this theoretical composition continued to claim the respect of teachers and composers late into the second half of the twelfth century.
A treatise by Guy, Abbot of Chalis, about this time, is concerned with essentially the same problems and presents no really new point of view. He is practically the last of the theorizing organizers. Organum gave way to a new kind of music. In the course of over two hundred years it had run perfectly within the narrow limits to which it had been inevitably confined, and the science of it was briefly this: to devise over any given melody a counter-melody which accompanied it note by note, moving, as far as possible, in contrary motion, sinking to meet the melody when it rose, rising away from it when it fell, and, with few exceptions, in strictest concord of octaves, fifth, fourths, and unison. Rules had been formulated to cover practically all combinations which could occur in the narrow scheme. The restricted, cramped art then crumbled into dust and disappeared. Again and again this process is repeated in the history of music. The essence of music, and, indeed, of any art, cannot be caught by rules and theories. The stricter the rules the more surely will music rebel and seek expression in new and natural forms. We cannot believe that music in the Middle Ages was not a means of expression, that it was not warm with life; and therefore we cannot believe that this dry organum of Hucbald and Odo, of Guido of Arezzo, of Guy of Chalis, which was still-born of scholastic theory, is representative of the actual practice of music, either in the church or among the people. On the other hand, these excellent old monks were pioneers in the science of polyphonic writing. Inadequate and confusing as their rules and theories may be, they are none the less the first rules and theories in the field, the first attempts to give to polyphony the dignity and regularity of Art.
Meanwhile, long before Guy of Chalis had written what may be taken as the final word on organum, the new art which was destined to supplant it was developing both in England and in France. Two little pieces, one Ut tuo propitiatus, the other Mira lege, miro modo, have survived from the first part of the twelfth century. Both are written in a freely moving style in which the use of concords and discords appears quite unrestricted. Moreover, the second of them is distinctly metrical, and in lively rhythm. It is noted with neumes on a staff and the rhythm is evident only through the words, for the neumes gave no indication of the length or shortness of the notes which they represented, but only their pitch. Now in both these little pieces there are places where the organizing voice sings more than one note to a note of the cantus firmus or vice versa. So long as composers set only metrical texts to music the rhythm of the verse easily determined the rhythm in which the shorter notes were to be sung over the longer; but the text of the mass was in unmetrical prose, and if composers, in setting this to music in more than one part, wished one part to sing several notes to the other’s one, they had no means of indicating the rhythm or measure in which these notes were to be sung. Hence it became necessary for them to invent a standard metrical measure and a system of notation whereby it could be indicated. Their efforts in this direction inaugurated the second period in the history of polyphonic music, which is known as the period of measured music, and which extends roughly from the first half of the twelfth century to the first quarter of the fourteenth, approximately from 1150 to 1325.
IV
Our information regarding the development of the new art of measured music comes mainly from treatises which appeared in the course of these two centuries. Among them the most important are the two earliest, Discantus positio vulgaris and De musica libellus, both anonymous and both belonging to the second half of the twelfth century; the De musica mensurabili positio of Jean de Garlandia, written about 1245; and at last the great Ars cantus mensurabilis, commonly attributed to Franco of Cologne, about whose identity there is little certainty, and the work of Walter Odington, the English mathematician, written about 1280, De speculatione musices. As the earlier theorists succeeded in compressing a certain kind of music within the strict limits of mathematical theory, so the mensuralists finally bound up music in an exact arbitrary system from which it was again to break free in the so-called Ars nova. But the field of their efforts was much larger than that of the organum and the results of their work consequently of more lasting importance.
The first attempts were toward the perfecting of a system of measuring music in time, and the outcome was the Perfect System, a thoroughly arbitrary and unnatural scheme of triple values. That the natural division of a musical note is into two halves scarcely needs an explanation. We therefore divide our whole notes into half notes, the halves into quarters, the quarters into eighths, and so forth. But the mensuralists divided the whole note into three parts or two unequal parts, and each of these into three more. The standard note was the longa. It was theoretically held to contain in itself the triple value of the perfect measure. Hence it was called the longa perfecta. The first subdivision of the longa in the perfect system was into three breves and of the breve into three semi-breves. But in those cases in which the longa was divided into two unequal parts one of these parts was still called a longa. This longa, however, was considered imperfect, and its imperfection was made up by a breve. So, too, the perfect breve could be divided into an imperfect and a semi-breve.
Let us now consider the signs by which these values were expressed. The sign for the longa, or long, as we shall henceforth call it, was a modification of one of the old neumes called a virga, written thus
