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 (breves) and shorter notes (semi-breves). These groups could be subdivided variously into groups of two and three, giving far more variety of rhythmic development. Furthermore, different voices could be under different mensurations at the same time, allowing some voices to move at slower rates and while some move at faster rates.
Figure 7: Below is a video and transcription of an isorhythmic motet from the "Mass of Tournai" from the 14th century. Notice how the lower voices is moving at a much slower pace than the middle voice, which itself is moving at a pace both faster than the lower voice but slower than the top voice. Franconian Notation allowed composers to notate the stratification of musical lines for the first time.
Idiomatic to this isorhythmic style was the stark independence of each musical stratum. Rather than all voices having equal rates of temporal progression, voices would be intentionally set at three different rates, or mensurations. As music would continue to develop into the 14th, 15th, and 16th centuries, this rhythmic stratification would continually decrease and return to relative equality, albeit with greater rhythmic variety than found in the 13th century Notre Dame School. While the early 15th century would see the popularity of Cantus Firmus masses and motets, which utilizes a similar form of rhythmic stratification, this would decline in popularity over this century, and ultimately loose favor in the 16th century, only resurfacing in the 17th and 18th centuries, in some, mostly German, contrapuntal organ and choral works, such as in Johann Pachelbel’s and Georg Bohm’s chorale fantasias for the organ and Johann Sebastian Bach’s later cantus firmus based organ works and vocal chorale fantasias, as found in his cantatas BWV-4 and BWV-80.
In the 20th century, such stark rhythmic stratification has returned prominently in the experimental music of the rhythmic avant-garde. Composers like Olivier Messiaen, in pieces like Visions de la Amen, and Conlon Nancarrow, in works like Study No. 17 for pl