This first crude draft is, as cautioned, a very partial introduction. I have focused on patterns rather than exceptions. The music itself presents patterns, variations, and exceptions - and any line between these categories must also necessarily be arbitrary.
As in most polyphonic music, there is a constant and often subtle interaction between the vertical and melodic dimensions. A rather transient tone may permit a smoother and more graceful melody at the same time as it results in a directed vertical progression. Almost any point where the lowest voice moves by step or by third provides an opportunity for "cadential action" in this freer sense, however momentary, and there are pieces where such progressions seem to occur at most transitions between measures in a modern score.
The sustained-tone organum passages of Perotin and his colleagues, like the "pedal harmonies" of other periods, present a special kind of vertical color. Typically such passages focus on a stable 8/5 or 8/4 sonority above the sustained tenor, but exploit the full range of combinations we have explored. At times the upper two or three voices may engage in directed progressions (Section 4) above or around the unchanging sustained note; a change in the tenor provides the opportunity for a decisive cadence involving all voices.
For an excellent example, see (and hear or perform) the end of the first portion of Perotin's Sederunt principles for four voices. To set the syllable "-runt", Perotin repeats a phrase twice. The first time, the vertical tension dissipates; the second time, it is released in the cadence with m7/5/m3 described in Section 4.4.
It is well to end on a note of humility: any modern explanation of this music must rest in good part on one's own artistic perception and conceptual imagination, hopefully not inconsistent with the theory of the period and the evidence of the music itself.
Notation graphics were prepared by José Rodríguez Alvira.
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Margo Schulter