Continental musicians adopted the cyclic Mass from the English as the standard "high" genre in the mid-the-early 1400's. As such, the Mass became the form composers were most drawn to to demonstrate their compositional prowess, something that had been happening since the Notre Dame school and can be found in the development of the isorhythmic motet, for example.
Each movement of a mass utilises the same "Cantus Firmus," and, seeing each movement's texts were drawn from disparate parts of the larger service, exemplifies the role literacy played in developing new forms in music; the mass' movements were combined only because of the common melody, not because of any over-arching textual narrative.
Around 1440, an anonymous English composer (possibly John Dunstable) created Missa Caput: a composition that reached over and above previous levels of grandeur. It is the first documented polyphonic composition in four parts. Scribes adopted a new way of naming the four voices: the tenor remained as such, while the parts directly above and below became known as contratenor altus and contratenor bassis respectively. The top-most part was known as the superius.
At cadences, the tenor and alto parts follow the convention of resolving, in contrary motion, from an imperfect consonance (a sixth or third) to a perfect consonance (octave or unison). The only way to have the two other voices form consonances with both the tenor and alto while remaining independent of them is to sing notes that result in two triads a fifth apart. Here, we have the first explicit example of the perfect cadence, V-I. Using a Landini sixth to, say, the soprano, would result in the fourth degree of the mode sounding, giving the V chord an added "seventh."
This should not be viewed as the "beginning" of tonal harmony; that is our tendency to see (early) examples as essences. However, it is a point where developments in form and the fusing of continental and English approaches to harmony helped give rise to occurrences we now regard as conventions.