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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

Like all previous notational reforms, the Ars Nova retained the familiar shapes of Gregorian “square” notation, modifying them where necessary (as in supplying the minim) but as slightly as possible. What mainly changed were the rules by which the signs were interpreted. The same notated maxima could contain 16 minims or 81 minims or any of several quantities in between. How was one to know which?

What was needed was a set of ancillary signs—time signatures, in short—to specify the mensural relationships that obtained between the notated shapes. Again, economy was the rule. These signs were adapted directly from “daily life”—that is, from existing measuring practices, particularly those involving time-measurement (chronaca) and “business math” (chiefly minutiae or fiscal fractions).3

In the fourteenth century, not only musical durations but weight, length, and the value of money were all measured according to the duodecimal (twelve-based) system inherited from the Romans, rather than the decimal (ten-based) system derived from counting on the fingers, only lately available in numerals borrowed from the Arabs. Roman weights and measures survived longest in Britain and its cultural colonies. In America, despite long pressure to convert to the decimal metric system, introduced as an “enlightened” by-product of the French Revolution, we still divide feet into twelve inches and pounds into sixteen ounces. (In Britain itself, the monetary system remained duodecimal until the 1970s, with twelve pence to the shilling, and 240 pence (12 × 20) to the pound.) Both “inch” and “ounce” are traceable to the Latin word uncia, which stood for the basic unit of duodecimal measurement, whether of weight, length, or money. The uncia was the equivalent, in those areas, of the basic unit of musical measurement, the tempus.

The standard Roman symbol for the uncial—on abacuses used for monetary transactions, for example—was the circle, and the symbol for one-half of an uncia (called the semuncia), logically enough, was the semicircle. It is hardly a coincidence, then, that the circle and semicircle were adopted as symbols for the division of the tempus (breves into semibreves) in Ars Nova notation, thus becoming the first standard time signatures used in Western music. The circle stood for tempus perfectum—i.e., the “whole” or “perfect” breve containing three semibreves—and the semicircle stood, correspondingly, for tempus imperfectum, with two semibreves to a breve.

The signs for major and minor prolation were adapted from the theory of chronaca, in which the shortest unit of time—sometimes called the atomus, sometimes the momentum, and sometimes, yes, the minima—was compared with the geometric point (punctum), defined by Euclid as that which cannot be subdivided. (“A point,” Euclid wrote in his Elements, “is that which has no part.”) The minimal time-unit was sometimes actually called the punctum, which is undoubtedly why the point, or dot, became the symbol for the musical minima and its mensuration. The major prolation, in which there were three minims to a semibreve, was at first indicated by placing three dots inside the circle or semicircle that represented the breve. The minor prolation was specified by a pair of dots.

Later on scribes figured out that they could save some ink by subtracting two dots from this scheme. Major prolation could just as well be indicated by a single dot, minor prolation by the absence of a dot. So by the end of the fourteenth century, the four tempus-cum-prolation combinations or meters listed above were represented by four standard time signatures: The last of them, the one that represented mensuration by two at all levels, still survives (as the sign for “common time”). In the light of the foregoing discussion, it should be obvious that explaining the “C” for meter as the initial of the expression “common time” is a folk-etymology. Its actual derivation was from medieval minutiae and chronaca, and its survival depended on its “imperfection.” The main difference between modern notation and mensural notation is that although we certainly have our modern ways of indicating triple meter, the whole ancient idea of triple or “perfect” mensuration has been shed.

Representing it

fig. 8-2 Ars Nova notation: the four signatures.

The table in Fig. 8-2 sums up the relationships specified by the mensural notation that was first employed by the Parisian musicians who promulgated the “Ars Nova” in the early fourteenth century, and these relationships remained the basis for musical notation in Europe almost to the end of the sixteenth century.


(3) See Anna Maria Busse Berger, Mensuration and Proportion Signs: Origin and Evolution (Oxford: Clarendon Press, 1993), Chap. 2.

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. 19 Jan. 2021. <https://www.oxfordwesternmusic.com/view/Volume1/actrade-9780195384819-div1-008004.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 19 Jan. 2021, from https://www.oxfordwesternmusic.com/view/Volume1/actrade-9780195384819-div1-008004.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 19 Jan. 2021, from https://www.oxfordwesternmusic.com/view/Volume1/actrade-9780195384819-div1-008004.xml