o more rapidly and assuredly unity people under one musical liturgy, this music should be written down such that the human mind would be less likely to accidentally "corrupt" (vary) it from region to region. Such needs soon brought a system of pitch notation to help insure the fidelity of the reproduced chant, no longer safeguarded only in the memories of cathedral choirmasters and cantors.
Consequently, the form of chant notation, often called “square notation,” evolved, particularly under the innovations of Guido of Arezzo, to effectively communicate the pitch content of any melodic line of music (see Figure 3). However, it did not communicate any aspect of rhythmic performance, if there was any aspect as such at all. 
Figure 2: Neumatic notation from the early Middle Ages. Gestural shapes written above the text to be sung. These shaped gestures were merely a reminder to the singer of the shape of the chant tune (the melody goes up, down, zig-zigs, etc.). This system predicated itself on the fact that the singer would already know the chant, having learned it from a teacher and memorized it. This system gave no precise indication of pitch or rhythm.
Figure 3: Square notation of chant, which now gives a clear reproduction of pitch content, the four lines and three spaces being a prototype of our modern system with one additional line. The placement of a clef (usually a moveable C-clef) at the beginning of the line would tell the singer the relation of half- and whole-steps on the staff of lines.
It wouldn’t be until the 12th and 13th centuries at the School of Notre Dame in Paris that musicians developed any system of written temporal demarcation. This system, often known as “the six rhythmic modes of Notre Dame,” would be rather limited, confined to only the various three-grouping permutations of “longs” and “shorts” (one time-unit and half that same unit, or like the modern quarter and eighth note), specifically reflecting the common meters in Latin poetry (see Figure 4).  This system of notation was based on square-notation, but embedded in its structuring were patters of neume groupings, which indicated the rhythmic mode )see Figure 5).
Figure 4: Below are the listing of the six rhythmic modes of the Notre Dame School of the 13th century. to the left one can see the neume-grouping patterns, which would indicate to the singer the rhythmic mode in which to sing the notes associated with the pattern's position on the staff. Notice that all these patters suggest compound time (groups of three 8th-notes) with only long and half-long (short) values as their constituent parts.
Figure 5: Below is the beginning of Perotin's "Viderunt Omnes" for Christmas (a famous example of Notre Dame polyphony). Notice that this notation of music looks highly similar on its surface to the notation of the monophonic chant from Figure 3. However, this music notation does communicate rhythm, whereas the chant from Figure 3 arguably does not. You can hear this piece in the video appended below this image.
Only by the end of the 13th century, was a system of high variability for rhythm devised. This was accomplished by music theorist, Franco of Cologne (c. 1280). Through his “Franconian Notation” and its use of hierarchical rhythmic mensuration, time could be complexly divided and subdivided into various strata, rather than only having polyphonic texture in the same temporal layer siting in one “mode” of that level or another. Through the creation of rhythmic notation and the subsequent creation of a hierarchical system of rhythmic expression, one could organize musical data like never before, creating numerous strata of musical information. Naturally, one of the first musical forms that developed from this innovation was predicated on such informational stratification: the isorhythmic motet (see Figure 6 and 7). 
Figure 6: The basic system of Franconian Mensural Notation. Given the mensural sign to the left (the prototype of our modern "time signature"), the performer could ascertain the hierarchy of rhythmic values given by longer notes (brev