In our age of utilitarian inspiration the imperfections of such a system of notation in which the two most frequent signs had a twofold significance would be remedied by the invention of other signs; but the theorists of that day found it easier and more natural to supplement the system with numbers of rules whereby the exact values of the notes could be determined. For example, a long followed by another long was perfect; a long followed by a breve was imperfect and to be valued as two beats. But a long followed by two breves was perfect, for the two breves in themselves made up a second perfect three, since one was considered as recta and the other as altera. A long followed by three breves was obviously perfect, since the three breves could not but make up a perfect measure. Similar rules governed the valuation of the breve. Three breves between two longs were not to be altered, four breves between two longs also remained unaltered, since one of them counted to make up the imperfection of the preceding long. But five breves required alteration, the first three counting as one perfect measure, the last two attaining perfection by the alteration of the second of them. Semi-breves were also subject to the laws of perfection and alteration and were governed by much the same laws as governed the breves. One who had mastered all these laws was able to read music with more or less certainty, though it must have been necessary for him to look ahead constantly, in order to estimate the value of the note actually before him.
Later theorists did not fail to associate the mysteries of the perfect system of triple values with the Trinity, and thus sprang up the belief that the earlier mensuralists had had the perfection of the Trinity in mind when they allotted to the perfect longa its measure of three values. Yet, clumsy as the system of triple values was, it was founded upon perfectly rational principles. It was the best compromise in music between several poetic metres, some of which, like the Iambic and Trochaic, are essentially triple; others, like the Dactylic and Anapæstic, essentially double. Music, during all the years while the mensuralists were supreme, was profoundly influenced by poetic metres. All these had been reduced by means of the triple proportion to six formulas or modes, and every piece of music was theoretically in one or another of these modes. Such a definite classification of various rhythms, besides being eminently gratifying to the learned theorists, was of considerable assistance to the singer in his way through the maze of mensural notation, who, knowing the mode in which he was to sing, had but to fit the notes before him into the persistent, generally unvarying, rhythm proper to that mode. Composers were well aware of the monotony of one rhythm long continued. They therefore interrupted the beats by pauses, and occasionally shifted in the midst of a piece from one mode to another. The pauses were represented by vertical lines across the staff, and the length of the pause was determined by the length of the line—the perfect pause of three beats being represented by a line drawn up through three spaces, the imperfect pause of two beats by one crossing two spaces and the others in proportion. The end was marked by a line drawn across the entire staff.
So far the complexities of the mensural system of notation are not too difficult to follow with comparative ease. But the longs, the breves and the semi-breves were employed only in the notation of syllabic music; that is, of music in which each note corresponds to a syllable of the text. In those cases where one syllable was extended through several notes, another form of notation was employed. The several notes so sung were bound together in one complex sign called a ligature. The ligatures, like the longs and the breves, were adaptations of old neumatic signs. In the old plain-song the flourishes or melismas on single syllables were sung in a free rhythm; but the mensuralists were determined to reduce every phrase of music to exact rhythmical proportions, and these easy, graceful, soaring ornaments were crushed with the rest in the iron grip of their system. Hence the ligatures were interpreted according to the strictest rules. A few examples will serve to show the extraordinary complexity of the system. Among the old neumatic signs which stood for a series of notes two were of especially frequent occurrence. These were the podatus,
, and the clivis,
. Of these the first represented an ascending series, the second—which seems to have developed from the circumflex accent—a descending series. It will be noticed that the clivis begins with an upward stroke to the first note, which is represented by the heavy part of the line at the top of the curve. The podatus has no such stroke. Several other signs were derived from these two, and those derived from the clivis began always with this upward stroke, and those from the podatus were without it. Thus all ascending ornaments were represented by a neume which had no preliminary stroke, all descending ornaments by one with the preliminary stroke. This characteristic peculiarity was maintained by the mensuralists in their ligatures. The podatus became
, the clivis
