Ars Nova theorists Philippe de Vitry and Jehan des Murs wrote treatises on the theoretical aspects of music. They sought to reconcile the rhythmic character of the twelfth-century motet, where a "longa" was equal to twice the "breve," and the thirteenth-century, "Franconian," motet, where the "longa" equalled a "perfection" of three "tempora."
From a mathematical perspective, the innovations that reconciled those rhythmic irregularities were the by-product of the theory of exponential powers and the theory of "harmonic numbers."
In basic terms, it was proposed that the breve could be broken into 3 (perfect, deriving from the holy trinity) or two (imperfect) semibreves. Semibreves could then also be broken into perfect or imperfect parts. There are four combinations of these:
1. Breve division: 3 Semi-breve division: 3
2. Breve division: 3 Semi-breve division: 2
3. Breve division: 2 Semi-breve division: 3
4. Breve division: 2 Semi-breve division: 2
The breve was roughly the equivalent of what we might now consider a bar. Therefore, combination 1 might represent 9/8 time, combination 2 3/4 time, combination 3 6/8 time,and combination 4 2/2 time.
One main difference between this and our notion of time signatures is that the above division do not dictate an accentuation scheme.