ayer piano, returned to the idea of measured or “mensural” rhythmic stratification (see Figure 8 and 9). Our modern system of musical notation, while allowing for such extreme rhythmic layering, does not suggest it obviously, generally requiring all instruments within a work to share the same time signature and barlines. Furthermore, most conventional uses of our modern system require barlines at all times, and these barlines should typically be relatively short, usually 2 to 7 beats long with 7 being rather exceptional. It is nothing new to note, but an understanding of a system of music that operates without barlines, under stratified rather than homogenous rhythmic hierarchies would allow for different modes of music composition, as demonstrated in the works of the two above composers as well as Karlheinz Stockhausen, John Cage, George Crumb, and many others. Many young music students, so arbitrarily entrenched in the modern metric system, are totally ignorant of these possibilities.
Figure 8: Messiaen's "Vision de la Amen" (see my post on this piece for further information on its isorhythmic structuring.
Figure 9: Nancarrow's "Study No. 17" for player piano, which has three musical strata in canon under the temporal proportions 12/15/20, realized as the three contrasting tempos of 138, 172.5. and 230 beats per minute. The three musical lines contain all the same pitch and rhythmic content (albeit different octave transpositions), however each voice is moving at a different rate according to the three above tempos.
The second profound development in Western music notation, which has perhaps given us more than it has taken, is the proliferation of “score” notation, that is, the notation of all polyphonic parts in music such that they all visually appear meaningfully juxtaposed (usually in a vertical list) according to their progression over time. Typically, this “meaningful juxtaposition” is the precise, concordant, periodic ordering of their rhythmic events from the shortest to the longest durations within any part. By developing this tool, composers were allowing themselves more complex methods to organize their ideas as well as communicate more information and more complex information to the performer, theorist, and eventually ensemble coordinator (conductor). Before score notation, it was more difficult, given only the independently notated polyphonic parts of a piece, to determine how one part would fit precisely with another and what the combination of these parts would ultimately communicate. With score notation, it became much easier to a priori internally audiate and understand a piece of music before it was ever unencrypted by performers in performance or rehearsal. (see Figure 10 and 11)
Figure 10: Below is the original manuscript of Alexander Agricola's (1457/58 - 1506) "Agnus Dei" from his "Missa in Myn Zyn." Here you can see the Soprano voice's part on the top of the left page, the "alto" (tenor 1) voice on the bottom of the left page, the "tenor" (tenor 2) on the top right, and the bass on the bottom right. While this edition is fine for performances purposes, it would be difficult to select any random note from one part and know immediately to which notes it corresponds in the other parts, since these parts are not temporally synchronized in this notation (e.g. the first note in the second line of the soprano does not necessarily correspond to the first note in the second line of any of the other parts).
Figure 11: Below is a video transcription with recorded performance of the above music manuscript. With the "score notation" it is much easier to see how each part harmonically and rhythmically fits together at any one moment.
There is no specific “light-bulb” moment for this innovation; no particular person is easily given the sole credit. Such notation in various forms appears in isolation throughout the Middle Ages and Renaissance, especially in theoretical treatises,  which perhaps realized the pedagogical utility of such a presentation of information. However, when only performance is the aim of mu