In the late fourteenth and early fifteenth centuries rhythmic proportions were indicated through coloration, Italian note shapes, mensuration signs, and fractions. Even though Johannes de Muris had introduced in 1321 a division of the breve into from two to nine equal parts, theorists and composers used only those proportions which could be indicated by a combination of various mensuration signs with the assumption of breve equivalence: 3:2 shown by ○:⊂ or ⪽:⊂; 4:3 by ⊃:○ or ⊃:⪽; 9:4 by ⊙:⊂; 9:8 by ⊙:⊄; 2:1 by ⊂:⊄; 8:3 by ⊄:⪽. Other proportions were not used because musicians lacked adequate signs to indicate them. The invention of the fraction to show rhythmic proportions presented, therefore, a true innovation because it permitted the indication of proportions not naturally inherent in the mensural system. However, fractions were used until the late fifteenth century as if they were mensuration signs, that is, they were not cumulative and they determined the mensuration of the following section. It was not until the late fifteenth century that Johannes Tinctoris and Franchinus Gaffurius emancipated proportions from mensuration signs and used fractions in an arithmetically correct way.
- Copyright 1988 The American Musicological Society, Inc.