other direction, so to speak), three perfect longs could be grouped within a perfect maxima or longa
triplex.
We are thus working within a fourfold perfect system expressible by the mathematical term 3^4 , “three
to the fourth,” or “the fourth power of three.” The minim is the unit value. Multiplied by 3 (3^1 ) it produces
the semibreve, which has three minims. Multiplied by 3 × 3 (3 ^2 ) it produces the breve, which has nine
minims. Multiplied by 3 × 3 × 3 (3^3 ) it produces the long, which has 27 minims; and multiplied by 3 × 3 ×
3 × 3 (3 ^4 ) it produces the maxima, which has 81 minims. Each of these powers of three constitutes a level
of musical time-division or rhythm. Taking the longest as primary, Jehan des Murs called the levels
- Maximodus (major mode), describing the division of the maxima into longs;
- Modus (mode), as in the “modal” rhythm of old, describing the division of longs into breves, or
tempora; - Tempus (time), describing the division of breves into semibreves; and
- Prolatio (Latin for “extension,” usually designated in English by an ad hoc cognate,
“prolation”) describing the division of semibreves into minims.
And he represented it all in a chart (Fig. 8-1) which gives the minim-content of every perfect note
value in “Ars Nova” notation.
FIG. 8-1 Harmonic proportions according to Jehan de Murs.
And now the stroke of genius: The whole array, involving the very same note values and written
symbols or graphemes, could be predicated on Garlandia’s “imperfect” long as well as Franco’s perfect
one, from which a fourfold imperfect system could be derived, expressible by the mathematical term 2 ^4 ,
“two to the fourth,” or “the fourth power of two.” Again taking the minim as the unit value, multiplied by 2
(2^1 ) it produces a semibreve that has two minims. Multiplied by 2 × 2 (2 ^2 ) it produces a breve that has
four minims. Multiplied by 2 × 2 × 2 (2 3) it produces a long that has 8 minims; and multiplied by 2 × 2 ×
2 × 2 (2 ^4 ) it produces a maxima with only 16 minims.
So at its perfect and imperfect extremes, the “Ars Nova” system posits a maximum notatable value
that could contain as many as 81 minimum values or as few as 16. But between these extremes many other
values were possible, because the levels of maximodus, modus, tempus, and prolatio were treated as
independent variables. Each of them could be either perfect or imperfect, yielding on the theoretic level
an exhaustive array of “harmonic numbers,” and, on the practical level, introducing at a stroke as wide a
range of conventional musical meters as musicians in the Western literate tradition would need until the
nineteenth century.