978-94-015-7686-4_3

1. CHAPTER 3 THE EXPERIMENTAL APPROACH In the preceding chapter we investigated the results of the mathematical approach to the problem of consonance as defined by Zarlino. In the hands of Kepler this new approach led to a geometrical theory of harmony, while it induced Stevin to deny that the consonances are defined by the first few integers at all. However idiosyncratic the consequences were that Stevin was willing to draw from his argument against the supposed excellence of the first few integers, these do not dirninish the basic soundness of his starting point. In terms oi number alone, the argument was valid. It lost its relevance, however, from the moment the phenomenon of consonance was placed on a new, physical basis. In fact, a first step in this direction had already been taken some forty years earlier. 3.1. GIOVANNI BATTISTA BENEDETTI Giovanni Battista Benedetti (1530-1590) is the prototypical forerunner, aperson, that is, whose work appears significant onIy in the light of that of a later one who did better. In Benedetti's case the successful successor was Galileo. As far as mechanics is concemed (projectile motion, free fall), this has been known for a long time.! In the case of music the discovery is much more recent. It was not until 1961 that Palisca called attention to a particular passage in one of two letters written around 1563 by Benedetti to the composer Cipriano de Rore. In 1585 Benedetti inserted the letters in his Diversarum speculationum mathematicarum et physicarum liber ('Book of Various Mathematical and Physical Ideas').2 The content of the letters makes it c1ear that Benedetti was quite know- ledgeable about music; in fact he appears to have been a composer hirnself. His second letter ends with abrief, 40-line theory on the generation of the consonances through the 'cotermination of percussions'. (A brief overview of the other subjects treated in the letters is to be found in Note 3.) Benedetti's discussion of consonance runs as follows. When the string of, for example, a monochord is plucked, it regularly strikes ('percusses') the surrounding air; the resulting air waves generate sound. When the string .is halved, both parts will emit equal-pitched sounds, because in the same time 75 H. F. Cohen, Quantifying Music © Springer Science+Business Media Dordrecht 1984

2. 76 CHAPTER 3 both will make the same number of percussions. Hence they constitute a unison. The air waves do not cut through each other, or break each other, but they concur entirely. When one part of the string is twice as long as the other, the resulting octave is obviously formed by percussions that have a ratio of 2: I, since in the time the longer part needs for one percussion the shorter one will have performed two. In other words, every percussion of the longer string coincides with every second percussion of the shorter one, "since there is no one who does not know that the longer the string, the slower its motion".4 In the case of the fifth, too, the number of vibrations per unit time is inversely proportional to the respective string lengths, hence their ratio is as 3: 2. Here coincidence will not occur until the shorter string has performed three vibrations, and the longer one two. As a result the product of the numbers for the length and the vibrations per unit time of the shorter string (2 X 3) will equal the product of the corresponding numbers for the longer string (3 X 2). Extending this result to all consonances, we get the fol1owing se ries ofproducts (Figure 34): [Unisonll octave IfIfth I fourth I major sixth I major third I minor third I minor sixth 1 2 6 12 15 20 30 40 Fig.34. which numbers correspond with each other in a wonderful proportion [non absque mirabili analogia]. Now the pleasure that the consonances give to hearing comes from their softening the senses, while, to the contrary, the pain that originates from the dissonances is born from sharpness, as you can easily see when organ pipes are tuned.s With these words the letter ends. In the light of al1 the ideas that later were to branch out from theories sirnilar to this one, it is easy to overestimate the importance of the theory in the form Benedetti gave it, and to read into it a meaning that is not yet there. Palisca observes that "Benedetti's discovery was potential1y a fatal blow to [Zarlino-style] number symbolism". 6 This is true only if the word 'potentially' is emphatically stressed. For apparently Benedetti's operations with the product of a string's length and the number of vibrations it makes per unit time do not differ so fundamentally from Zarlino's mode of thought as Palisca, would have it. According to Palisca, Benedetti multiplies these numbers in order to draw up a table in which degrees of consonance are compared (as was to be done later by Mersenne; see Section 3.5.3.). But this aim is not stated anywhere. In fact, Benedetti's only concern was to