listening ear but can be easily grasped and relished by the rational mind. This aspect of the Mass, in other
words, belonged to the level not of “music” as we understand the term, but of Musica, as understood in
the enduring tradition of Boethius, first described in chapter 3.
When the durations of every subsection of the Mass are measured against the initial tempus or
counting unit established in the first Kyrie, a breathtaking array of “Pythagorean” proportions is revealed
(Fig. 12-9). It resembles the one illustrated in Ex. 12-1a, from Busnoys’s motet In hydraulis; but it is
more hidden, farther-reaching, and far more complete. To pick one example: as shown in Fig. 12-9, the
four written subsections of the Sanctus contain 36, 27, 18, and 24 tempora respectively. The ratio
36:27:18:24 reduces (when divided by three) to 12:9:6:8—exactly the proportions of the anvil-weights in
the old story of Pythagoras and the blacksmith’s shop, an array that yields all the Pythagorean consonances
(see Ex. 1-9): the octave (12:6 = 2:1), the fifth (9:6 = 3:2), the fourth (8:6 = 4:3), and even the “tone” or
major second, the difference between the fourth and the fifth (9:8).
The concluding Agnus Dei has three subsections in the durational proportion 36:27:18, which when
factored by nine reduces to 4:3:2, an array that precisely and economically sums up the perfect
consonances, just like the final octave/fifth/fourth harmony of this or any fifteenth-century Mass or motet.
The subsections of the Kyrie have lengths can be represented as the ratio 18:16:18, reducing by a factor
of two to 9:8:9, expressing the tone. Also striking is the fact that the opening sections of each of the five
“movements” in the Mass collectively make an array that reduces to 1:3:3:2:2, expressing the most basic
consonances, the octave and the fifth; while the concluding sections of each “movement” have identical
durations (18 tempora), thus collectively expressing absolute unity.
FIG. 12-9 Durational proportions in Antoine Busnoys’s Missa L’Homme Armé.
Right smack in the middle of things, a prime number occurs in the durational plan that seems to skew
it. But that number is 31, symbolizing the Order of the Golden Fleece. So far from skewing the plan, the
existence of 31 as a durational unit provides further evidence that Bus noys attached symbolic
significance to durations and planned them out in advance, just as a composer of ceremonial motets might
formerly have done.
It is worth reiterating that this impressive numerological edifice cannot be heard in performance, nor
can it have been meant to be heard. It is not even possible to sing the Mass with the ideal tempo
proportions that would realize the ground plan accurately, for that would put the sections in diminution
beyond the likely abilities of even the most agile singers. And that is precisely the point. Musica (as