167th Acoustical Society of America Meeting

Marina Baldissera Pacchetti – mab360@pitt.edu

History and Philosophy of Science

University of Pittsburgh

4200 Fifth Avenue

Pittsburgh, PA

15213

Popular version of paper 1pMU1

Presented on Monday afternoon, May 5, 2014

167th ASA Meeting, Providence

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The history of science allows us to gain new perspectives and give new interpretations to the outlooks we have on the scientific practice today. In my paper, I have investigated a little-explored part of the history of science, namely the history of acoustics. The start of modern acoustics is usually marked by the investigation of the Parisian monk Marin Mersenne. However, I claim that this is to be attributed to the collaboration and disagreement of three scholars that worked almost a century earlier, the most notable of which is Vincenzo Galilei, Galileo's father. The documents I have worked on have been collected by the philologist and musicologist Claude Palisca (1921-2001).

The period I investigated, the Renaissance (16th century), is marked by the revival of Platonic and Pythagorean traditions that favoured reason over sense perception, the latter being considered misleading. This was in stark tension with the way Aristotle was read at that time, namely as favouring an empirical and sense driven approach to the study of nature. During this period, scholarship on music theory undergoes, I believe, an important change.

Giovanni Bardi (1534-1612) was a Florentine nobleman interested in the Arts. He started what one might call a prototype of the later formal "Accademia," called the "Camerata de' Bardi." The Camerata allowed for intellectuals to meet and exchange lectures and ideas on a regular basis, led by Bardi himself.

At that time, music theory was an important discipline, studied in the *Quadrivium* that, together with the *Trivium,* determined the subjects to be studied by what we would call today an undergraduate student. The Quadrivium consisted of the mathematical sciences: arithmetic, geometry, music and astronomy. The fact that music was part of the quadrivium is a sign that music was an important, recognized theoretical discipline and was part of what we would call today the "mathematical sciences." One has to note that at this point in history the use of mathematics to study what we call "physics" today was not yet well defined. Scholars still debate, and there is much to be debated still, about what the role of mathematics was in ancient Greece, in the Arabic Empire and in the Middle Ages (and today, for that matter).

Let us however glide over this issue for now. What is important is that, at least in the study of music, there is a strong divide between theoretical and practical applications of music theory. The theoretical approach relied heavily on what is known in the literature as the "Pythagorean doctrine," which postulated that pre-established mathematical ratios were supposed to determine which combination of sounds were consonant or not. The demonstrations that were invoked were geometrical proofs, and the ratios were all verified on a single string, or two strings of equal tension. This doctrine was preserved in the Western tradition by Boethius (6th century AD), by Pietro D'Abano (1260-ca.1316) and later Franchino Gafurio (1451-1522). Gafurio is an interesting character because he was the first to note that the Pythagorean ratios (1:2, 3:2, 4:3 and particular combinations thereof) did not really sound "pleasant" when played on an instrument tuned according to the "equal temperament," a tuning used by most lutenists at the time.

By the time that Bardi was running his Camerata, the most prominent scholars that dealt with music theory, and indirectly also acoustics, were Gioseffo Zarlino (1517-1591) and Girolamo Mei (1519-1594). These were major influences on Vincenzo Galilei (1520-1591) who would perform experiments that disproved the Pythagorean doctrine of a priori defined musical ratios.

Vincenzo was a renowned lutenist, but had little theoretical training. After being recruited by Giovanni Bardi, Vincenzo was sent to study with the eminent music theorist Gioseffo Zarlino in Padua, just while Zarlino was composing his masterpiece, *Le Istitutioni Harmoniche,* that celebrated the Pythagorean tradition according to which only a specific set of numbers defined *a priori* (i.e. without empirical evidence) could explain consonant numbers. Once returned to Florence, however, Vincenzo was trying to interpret the Greek sources that inspired Zarlino, most notably the work of the polymath Ptolemy (2nd century AD), who wrote the famous Almagest, which described the geometry of the universe. This is indicative of what music theory was at the time: a set of theorems, mainly of geometry, from which then the theorist would extrapolate the results that applied to music. Vincenzo also started reading the work of another important Greek music theorist, Aristoxenus (3rd century BC), who, on the other hand, claimed that mathematics had nothing to do with music, and that the senses were the best arbiter for the attributes of musical sound. Confused by these contradicting statements, Vincenzo turned to Girolamo Mei, who was a philologist whose main interest was finding and translating ancient Greek texts on music theory. He explained to Vincenzo that contemporary music theorists like Zarlino were making a mistake in applying strict geometrical rules to music. Mei further explained to Vincenzo that the "equal temperament," which was the one that sounded most pleasant to Western Renaissance ears, was indeed the one of Aristoxenus, and therefore did not conform to the Pythagorean doctrine that particular ratios *determined* particular consonant intervals. Mei encouraged Vincenzo to verify for himself: he should tune a lute according to the tuning of the theoreticians, and another lute to the equal temperament of the practitioners, that corresponded to Aristoxenus's tuning, and then verify which intervals were the consonant ones. The problem that arose is that the theoretical tuning would recover all the intervals precisely, but it was not the one used at the time. Further, not all the intervals considered consonant at Vincenzo's time could be recovered. On the other hand, the equal temperament, did not recover most of the ratios imposed by the Pythagoreans. This indicated that *a priori* mathematical arguments could not be applied arbitrarily to sound, which indicated that the theory was faulty.

Vincenzo, therefore, came up with what we would now call an "experiment" (though the word he used was "experience") to show that mathematical ratios do not determine what is consonant, but that different physical systems that can be related by different quantities can produce consonant sounds. The consonance is ultimately to be established by the judgment of the ear. The experiment was very simple, and it will be enough to explain only part of it: Vincenzo showed that the same kind of consonance, the octave, could be produced by applying different ratios to different physical characteristics of the instrument. The ratio 1:2 only applied to strings of constant tension: if two strings have the same tension, and one is half the length of the other, then they will produce a consonant sound called the octave. However the same sound can be produced if we have two strings of the same length, but are tensed by weights whose weights yield the ratio 1:4. He could therefore empirically show that consonances *were not a priori defined by ratios*, but the ear would judge what sounded consonant, and empirical investigation would allow one to quantify the relevant variables of the systems that did produce the consonance. Vincenzo described the experience in detail, such that anyone could perform it and verify for themselves.