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Music from the Earliest Notations to the Sixteenth Century


CHAPTER 8 Business Math, Politics, and Paradise: The Ars Nova
Richard Taruskin

From a purely mathematical point of view, the Ars Nova innovations were a by-product of the theory of exponential powers and one of its subtopics, the theory of “harmonic numbers.” It was in the fourteenth century that mathematicians began investigating powers beyond those that could be demonstrated by the simple geometry of squares and cubes. The leader in this field, and one of the century’s leading mathematicians, was Nicole d’Oresme (d. 1382), the first French translator of Aristotle, whose writings (as we have already seen) encompassed music theory as well. His career as scholastic and churchman closely paralleled that of Philippe de Vitry: Philippe ended his ecclesiastical career as the Bishop of Meaux, northeast of Paris; Nicole ended his as Bishop of Lisieux, northwest of Paris. Nicole d’Oresme’s Algorismus proportionum was the great theoretical exposition of fourteenth-century work in “power development” (recursive multiplication) with integral and fractional exponents; but it was precisely in Jehan des Murs’s music treatise that the fourth power first found a practical application.

As for “harmonic numbers,” this was a term coined by the mathematician Levi ben Gershom (alias Gersonides or Leo Hebraeus, 1288–1344), a Jewish scholar who lived under the protection of the papal court at Avignon. Gersonides’s treatise De numeris harmonicis was actually written at the request of Philippe de Vitry and partly in collaboration with him. It consists of a theoretical account of all possible products of the squaring number (2) and the cubing number (3), and their powers in any combination.

All of this became music, first of all, in the process of rationalizing the “irrational” divisions of the breve into semibreves, with which, as we saw at the end of the previous chapter, composers like Petrus de Cruce had been experimenting at the end of the thirteenth century. And the other “problem” that motivated the Ars Nova innovations was that of reconciling the original twelfth-century “modal” concept of the longa as equaling twice a breve (that is, the two-tempora long of “Leonine” practice as later codified by Johannes de Garlandia) with the thirteenth-century “Franconian” concept of the longa as equaling a “perfection” of three tempora.

In turn-of-the-century “Petronian” motets, like Ex. 7-10, a breve could be divided into anywhere from two to nine semibreves. The obvious way of resolving this ambiguity was to extend the idea of perfection to the semibreve. The shortest Petronian semibreve (1/9 of a breve) could be thought of as an additional—minimal—level of time-division, for which the obvious term would be a minima (in English, a “minim”), denoted by a semibreve with a tail, thus: . Nine minimae or minims would thus equal three perfect semibreves, which in turn would equal a perfect breve. All of this merely carried out at higher levels of division the well-established concept of ternary “perfection,” as first expressed in the relationship of the breve to the long. On a further analogy to the perfect division of the long (but in the 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 34, “three to the fourth,” or “the fourth power of three.” The minim is the unit value. Multiplied by 3 (31) it produces the semibreve, which has three minims. Multiplied by 3 × 3 (32) it produces the breve, which has nine minims. Multiplied by 3 × 3 × 3 (33) it produces the long, which has 27 minims; and multiplied by 3 × 3 × 3 × 3 (34) 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

  1. 1. Maximodus (major mode), describing the division of the maxima into longs;

  2. 2. Modus (mode), as in the “modal” rhythm of old, describing the division of longs into breves, or tempora;

  3. 3. Tempus (time), describing the division of breves into semibreves; and

  4. 4. 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.

Music from Mathematics

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 24, “two to the fourth,” or “the fourth power of two.” Again taking the minim as the unit value, multiplied by 2 (21) it produces a semibreve that has two minims. Multiplied by 2 × 2 (22) it produces a breve that has four minims. Multiplied by 2 × 2 × 2 (23) it produces a long that has 8 minims; and multiplied by 2 × 2 × 2 × 2 (24) 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.

To deal, briefly, with the speculative side (since it was that side that initially drove the engine of change), maximae could now contain the following numbers of minimae between the extremes we have already established:

[High end (all perfect) 3 × 3 × 3 × 3 (34) = 81 minimae]

Any one level imperfect 3 × 3 × 3 × 2 (33 × 21) = 54 minimae

Any two levels imperfect 3 × 3 × 2 × 2 (32 × 22) = 36 minimae

Any three levels imperfect 3 × 2 × 2 × 2 (31 × 23) = 24 minimae

[Low end (all imperfect) 2 × 2 × 2 × 2 (24) = 16 minimae]

By similar calculations one can demonstrate that the long can contain 27, 18, 12, or 8 minims; a breve can contain 9, 6, or 4 minims; and a semibreve can contain 3 or 2 minims. The array of all numbers generated in this way, beginning with the unit—1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 54, 81—is the array of what Gersonides called harmonic numbers, since they are numbers that represent single measurable durations that can be fitted together (“harmonized”) to create music.

Citation (MLA):
Richard Taruskin. "Chapter 8 Business Math, Politics, and Paradise: The Ars Nova." The Oxford History of Western Music. Oxford University Press. New York, USA. n.d. Web. 7 Oct. 2022. <https://www.oxfordwesternmusic.com/view/Volume1/actrade-9780195384819-div1-008002.xml>.
Citation (APA):
Taruskin, R. (n.d.). Chapter 8 Business Math, Politics, and Paradise: The Ars Nova. In Oxford University Press, Music from the Earliest Notations to the Sixteenth Century. New York, USA. Retrieved 7 Oct. 2022, from https://www.oxfordwesternmusic.com/view/Volume1/actrade-9780195384819-div1-008002.xml
Citation (Chicago):
Richard Taruskin. "Chapter 8 Business Math, Politics, and Paradise: The Ars Nova." In Music from the Earliest Notations to the Sixteenth Century, Oxford University Press. (New York, USA, n.d.). Retrieved 7 Oct. 2022, from https://www.oxfordwesternmusic.com/view/Volume1/actrade-9780195384819-div1-008002.xml