13th-century composer, theorist, and scholar, Petrus de Cruce was apparently born in or near Amiens, in north-central France; for dates we know only that he was active in the years around 1290.
The beginning of such a solution was Franconian notation, so called after the theorist Franco of Cologne, who outlined the system in his c. 1260 treatise, Ars cantus mensurabilis (The art of mensurable music).
There was no provision in this notation for equal duple division, which (along with imperfect tempus, therefore) would have to wait until de Vitry codified the concept of prolation in his Ars nova of 1322.
Since composers wanted to maintain speech rhythms in their tripla, they looked for a way to divide the tempus into more than three semi-breves, which in modern notation would be the equivalent of a tuplet (triplet, quartolet, quintuplet, etc.
Also, more notes and more intricate subdivision led to a slowing of general tempo—the semi-breve was performed more slowly than it had been in earlier practice,[citation needed] becoming the true unit of the beat, and the lower voices lost their rhythmic vitality, becoming mere structural successions of breves and longs.