During the Renaissance there was no science of sound as we currently understand it. Conjectures about sound were provided by observations of particular acoustical phenomena incorporated in buildings or in open spaces, like whispering galleries or echoes as discussed in architecture. Sound was also a topic of interest for those who studied the functioning of natural bodies (i.e., voice and hearing), artificial instruments and machines (e.g., automata and war machines), among others. Although sound could not be identified as the subject matter of any science, it was attributed a relevant role in music treatises. It is assumed that the philosophical traditions involved in its study approached sound either as motion or as the object of hearing. However, up to the seventeenth century sound, as part of music, was mostly explored within the Aristotelian framework of subalternate sciences, being studied not as a perceived quality but as the natural quality of number, the actual object of music.
KeywordsSixteenth Century Musical Instrument Mathematical Science Natural Philosopher Music Theory
Many were the Renaissance authors that approached sound in their writings. Some of these authors attributed more relevance to historical narratives about its origin as represented by the character of Juval in the Bible, book of Genesis, and his invention of consonance; Pythagoras and the invention of consonance; as well as Hugh of St. Victor and the origin of singing.
Narratives about the invention of sound as the invention of musical consonance (agreeable musical intervals) was a recurrent theme found already in medieval treatises up to the late sixteenth century (Bromberg and Alfonso-Goldfarb 2010). The invention of consonance by either Pythagoras or Juval was transmitted as told by Iamblicus, Nicomachus of Gerasa, and Theon of Smyrna. Such legend was specially well illustrated in the works of Franchino Gaffurio (1451–1522), a music theorist from the fifteenth century. The legend explained that instruments as different as auloi, panpipes, and vessels filled to various levels with liquid were showed to yield the same proportional ratios as monochords when subjected to different tensions by attaching weights to them (Gaffurio 1993).
Some Renaissance authors mentioned whispering galleries and echoes as types of sound phenomenon, others were interested in the organ of hearing, and others yet spoke of a certain sound in the cosmos, as was the case with the theory of the harmony of the spheres in which sound was emitted through the movements of the celestial bodies while remaining unheard.
Whispering galleries are a good example of the intersection between building and natural philosophy. Some authors would assert that they had been planned in advance, while others would state that their acoustical phenomena were inherent to the construction and geometric form of the building that contained them. People could experience the same phenomenon outdoors as well, along the circular walls of certain gardens or in fortifications and also in some ancient theaters (Crunelle 1991). It is assumed that a truly geometrical and not an acoustical decision often guided the elaboration of the plans for the aforementioned buildings. The German Jesuit scholar Athanasius Kircher (1608–1680) described this effect in his work Musurgia Universalis (1620), a treatise written in a dispute with an English engineer. A second treatise called Phonurgia would elaborate on his interest in echoes, which he considered to be founded on sound waves propagated after hitting some obstacle (Tronchin et al. 2008). Nevertheless, the cause of such effects was attributed to geometry. Other natural philosophers, such as Giovanni B. Della Porta (1535–1615), also proposed a geometrical approach to acoustics when analyzing the powerful influence of music on human soul (Barbieri 2004).
Sound was something produced by an external body and was differentiated from voice. Voice was regarded as natural, while the production of other types of sound was considered to be artificial. Whether favoring or challenging Aristotle’s division between natural (voice) and artificial sounds (instruments, machines, etc.), authors attempted to recreate sound in automata, speaking machines, and musical instruments. Following Aristotle’s notion of perception, sound was also understood as the object of hearing. Authors would approach hearing as a sensory science, like sight; the analogy between sound and light was common among renaissance authors. Natural philosophers such as Francis Bacon, Athanasius Kircher, or Robert Hooke saw in acoustics the way to improve the range of hearing, or to improve communication (Gouk 1982, 2006), Galileo Galilei envisaged a new instrument aiming at bringing to bring sound closer (Valleriani 2012).
Regarding physical theories, it is assumed that the then most prominent philosophy of sound was that sound consisted of a kind of motion, a philosophy inherited from the Middle Ages. The idea that motion is generated by a source, which appeared in the works of Aristotle, was mostly represented in narratives comparing the propagation of sound with the water wave (Johnstone 2013). However, the wave analogy was not unanimously accepted; sometimes authors would not reveal preference for any theory, at other times they would make use of more than one. They admitted that sound could be propagated in a diversity of forms: wave, the scholastic notion of species, corpuscular theory, and atoms. The historiography, however, places the first solution for a sound theory at the moment when discussions about numbers were transformed into questions about vibrational frequencies (Dostrovsky 1975; Cohen 1984). In fact, the vibration theory would be regarded as the first theory of sound in the seventeenth century (Truesdell 1960; Dostrovsky 1975).
Regarding historical documentation, studies about sound have been preserved in music treatises since the Middle Ages. Medieval and Renaissance music was not classified as art but as science based on speculative knowledge and characterized by its mathematical rather than physical nature. Sound was a facet of discussions involving the definitions and demonstrations of music as a mathematical science, yet, in many cases, the perceived acoustical phenomena were also described.
Since sound was placed within music, it was in music treatises that definitions of sound appear. Music was based on mathematics, more precisely on the arithmetic of pitch. The arithmetic of pitch meant that sound was defined on arithmetical bases and treated by natural philosophers and theoretical musicians through arithmetical or geometrical analyses (Dostrovsky 1975; Szabó 1978). The arithmetical definition of musical elements was an inherited Pythagorean approach that dominated music theory throughout the Middle Ages and Renaissance. The mathematical definition of music was a consensus; Plato, Aristotle, and their followers had agreed on it, even though not in the same way (Truesdell 1976; Walker 1978; Cohen 1984; Barker 2007; Creese 2010).
Sound was analyzed solely through differences in pitch – the subject property that enables sounds to be compared with each other in terms of high and low. Aristotle argued that differences in pitch corresponded to differences in the motion of air, thus a low sound would correspond to a slow motion and high sound to a fast motion. No other attribute of sound was correctly identifiable, for timbre could not be completely differentiated from pitch, intensity was confused with pitch and timbre, and because there was no way to measure it outside the realm of music.
Questions about duration, intensity, or timbre were not pressing. The duration of sound was understood to be part of the studies about rhythm, which dealt with identifying time changes and managing the metrics of speech.
Nevertheless, the relationships between differences in pitch (high and low) were understood in terms of proportion. Musical intervals were considered multitudes in relation to one another and were expressible and measured only through integral number ratios. The specific qualification (consonant/ dissonant) and quantification (duple, triple, etc.) of ratios was dictated by textual tradition regardless of practice and experimentation.
Music belonged with arithmetic, geometry, and astronomy since the medieval quadrivium, a classification that aimed to define the content and methods of each discipline and to establish the relationships between them (Dyer 2007). The proper order of these mathematical sciences was discussed during the Middle Ages, and two classifications eventually became relevant to music. One was based on Severius N. Boethius’ (c. 480–524) De institutione arithmetica and De institutione musica and the second on Martianus Capella’s De nuptiis Philologiae et Mercuri. For the first author, music was considered to be a mathematical-arithmetical science. Boethius held arithmetic on the highest position among the four sciences (Masi 1983); for Capella, music, although still considered as a mathematical science, was paired off with astronomy (Masi 1983; Teeuwen 2002). Both treatises were largely commented and received glosses of speculative and philosophical characters. From these glosses it becomes clear that the disciplines from the quadrivium were based on the same notions of mathematical ratios (Teeuwen 2002).
Boethius’ classification became strongly widespread in music theory during the Renaissance (Barbera 1980). Like Aristotle, he defined scientific knowledge as a doctrine built on first principles and strict demonstration. In his system of sciences, Aristotle had defined music as a speculative mathematical science, yet a science of proportions that possessed a physical component (Aristotle 1984). He had a place for these sciences that were partly mathematical and partly natural, such as optics and music. Music was defined as the most natural of the mathematical sciences (Physics 194a7). This view, unknown to the early Middle Ages, was of great relevance to the development of the scientiae mediae, specially by Thomas Aquinas (1225–1274).
Regarding the subject of music, that ambivalence meant that number was its essential attribute, but also that it held an improper part, an accident. This accident or quality was sound which came from physics (Metaph. 987b27). Authors from the thirteenth to the sixteenth centuries embraced this definition, and expressions such as numerus sonorus, numerus relatus ad sonos were used in relation to the subject of music by Robert Grosseteste, Albert the Great, and Robert Kilwardby, for example (Rico 2005). Nevertheless, regarding the developments in acoustics, it was with Aquinas that the configuration of music as a subalternate science became more fruitful. He considered the subject of music as a composite entity with a mathematical form and a physical matter, and, by doing so, he adhered to Aristotle’s explanation of number as the formal cause of musical consonance. To Aquinas the scientie medie reach conclusions about sensible matter even though they proceed through mathematical principles.
Robert Grosseteste (1175–1253) also elaborated on the subject matter of arithmetical music. Following the definitions of species of numbers as proposed by Boethius and in accordance with the quadrivium, Grosseteste introduced the numerus relatus. The term was introduced in his commentary on the Posterior Analytics. In the Posterior Analytics, Aristotle assumes that sounds (musical intervals) acquire their particular qualities through their corresponding ratios; however, because Aristotle knew that there was no absolute consensus on this issue, he decided not to carry out an investigation on types of intervals and ratios but, instead, explained that they could be defined in terms of priority of the mathematical over the acoustical elements or vice versa. He therefore called mathematical harmonicists “those who investigate harmonics according to numbers” (Top. 107a15–16) and hearing-based harmonicists “those who knew the fact that” (An. Post. 79a2–4). Only mathematical harmonicists were in possession of demonstrations of the causes (An. Post. 79a2–4) (Creese 2010). When commenting on this difference, Grosseteste associated the mathematical harmonicists with numerus relatus and the hearing-based harmonicists with numerus relatus sonorus. Numerus relatus concerned all proportioned beings (McInerny 1990).
Authors from the fifteenth to sixteenth centuries drew on the views of Aquinas and Grosseteste, now trying to distinguish between ratio, sound, and sounding body. Sounding bodies were not the same as musical instruments; they were elements that produced sound and did not correspond to the whole musical instrument, which involved a much more complex mechanism. Strings, glasses, and tubes were considered sounding bodies.
Understanding Sounding Bodies
Sounding body is a term that appears in Renaissance treatises to name things that cause sound with pitch and therefore fall within musical mathematics. As we saw, theorists were familiar with the Pythagorean doctrine of musical ratios, but to understand the relationship between musical ratios and sound required to know how to attribute magnitude to sound (Saito and Bromberg 2015). Since magnitude enables a qualitative and quantitative description of concepts through numbers, and given that, (a) magnitudes can be reduced to numbers and (b) measuring involves comparison, the magnitude to be measured must be material (Saito-Bromb 2015). In music the most important method to render sounds comparable was to make them visible (Creese 2010). This process was achieved through an instrument called monochord, which consisted of a single string stretched over a soundbox, with two fixed bridges, one at either end, and a movable bridge that allowed the string to be divided at any point in between. Under the string were some scales that functioned like a rod or ruler (Greek kanōn).
This instrument was known since ancient times, although its function and meaning had changed over time. Again, the medieval theory of the monochord transmitted to the Renaissance was based on Boethius’ writings. The medieval monochord worked as a pitch-producing instrument for singers and appeared in writings aimed at the construction of organs and bells. By the time of the Renaissance, the monochord was still used by singers but was mostly referred to by authors whose works dealt with the construction and tuning of musical instruments (Adkins 1967). The monochord use could be related to the theory of proportions and would dispense with a theory of sound as movement or impact, for it would not enable their measurement.
The behavior of strings or tubes was not known to theorists. Luthiers, that is, instrument makers, would be consulted for information about it. As we saw, there was no theory that could properly account for the acoustic properties of either strings or tubes.
Information on strings and tubes could be found mostly in those works dedicated to the arts of mining, smelting, and metalwork. Vanoccio Biringuccio’s Pirotechnia (1540) is an example of a technical treatise that included details about organ pipes and a description of the “bell scale” (Biringuccio 1966). Descriptions of instruments with strings of varying types or thickness and varying pipe length, diameter, and volume clearly show how unusual were his distinctions between pitch and number, in particular, the distinction between pitch production and his definitions of qualities applied to the notion of timbre.
In the first treatise dedicated to musical organology Musica getutscht (1511), Sebastian Virdung considered three categories of musical instruments. In the third group, which he called instruments that resound (and that is generically translated as percussion), he included the hammers on the anvil from the legend of Pythagoras, bells and chime bells along with organ pipes, explaining that the functioning and measuring of these instruments followed the proportional theory described by Boethius.
Nevertheless, many treatises were written that described old and new instruments (Vincenzo Galilei, Lodovico Fogliano, Michael Praetorius, Girolamo Cardano, Ercole Bottrigari). In such treatises, even though illustrations and information on the instruments would be given, the knowledge of how to make them would actually not be provided, and there were a number of reasons for that. Sometimes the knowledge would be kept within the circles of luthiers, unavailable for disclosure. But most often authors were either still attached to textual tradition or unable to legitimize their own theories.
Acoustics and Mathematics
Between the fourteenth and sixteenth centuries the acoustical dilemma lied in reconciling the natural argument with the principles of mathematics philosophically embedded in the discussion of subalternate sciences and the possibility of metabasis (Aristotle). In the specific case of music, arithmetic, with its application to the ratios of musical intervals, had to cohere with geometry and its application to the extensions of sounding bodies. Yet, being arithmetic and geometry two different sciences, each one with its own methods, and sound not being measurable as a physical entity, no philosophy or theory of sound enabled analyzing the arithmetics of interval, the geometry of the sounding body, and the reality of sound.
By that time textual tradition had legitimated arithmetic as the prime science. Then, because of arithmetic’s concern with number (theoretical abstraction of the musical interval), which is discrete, and geometry’s and physics’ similar concern with matter (sounding body), which is continuous, the content of the former was given greater certitude than the content of the latter. Therefore, either the sounding bodies or sound itself would still take a long period of time to become a proper object of inquiry. The theory of proportions also gave an answer to perception. Since agreeable sounds were defined by rational numbers, there was no philosophy of sense-perception beyond the distinction between consonance and dissonance.
The mathematical focus of this discussion is evident from the sources cited in the majority of the documents. The separation of the general magnitudes of geometry from the discrete multitudes of arithmetic was provided in books V and VII of Euclid’s Elements. The authors of the fifteenth and sixteenth centuries discussed mostly book VII, which contains the demonstration of a purely numerical theory of magnitudes (Murdoch 1963). These authors focused on Aristotle’s Physics and Posterior Analytics, since the subject of natural science is analyzed especially in the second book of Physics, which explains how it differs from the object of mathematics and enumerates the types of causes used in its demonstrations.
The need to understand the application of abstract ratios to sounding bodies as relations between parts and of parts to the whole set voice apart, for the voice organ could not be subjected to either partition or measurement. It was probably when authors disconnected the practices of the division of the monochord from the behavior of strings that they managed to develop new approaches to sound. The development of the laws for the natural frequencies of vibrating strings would be relevant in the works of Galileo Galilei (1564–1642), whose Mathematical Discourses Concerning Two New Sciences (1638) contained the statement and discussion about frequency equivalence (Dostrovsky 1975) and Marin Mersenne (1588–1648). Nevertheless, prior to the end of the eighteenth century music remained classified as a mathematical science. Even natural philosophers from the seventeenth to early eighteenth centuries considered the phenomena of sound in connection with music. In the Two New Sciences, Galileo still referred to the questions of sound that included vibration, as problems of music.
The transition from the scientific analysis of music in terms of number to an approach with an essentially physical basis is well discussed in literature (Gozza 2000; Cohen 1984) in which it is described all the major natural philosophers from the seventeenth century who devoted themselves also to music as was the case of Johannes Kepler, Simon Stevin, Francis Bacon, Renè Descartes, Pierre Gassendi, and Isaac Beeckman.
It is assumed that the mathematical theory of sound propagation began with Isaac Newton (1642–1727), whose Principia (1686) included a mechanical interpretation of sound as being “pressure” pulses transmitted through neighboring fluid particles. Substantial progress towards the development of a viable theory of sound propagation resting on firmer mathematical and physical concepts was made during the eighteenth century by Euler (1707–1783), Lagrange (1736–1813), and d’Alembert (1717–1783).
- Aristotle. 1984. In The complete works of Aristotle, ed. J. Barnes. Princeton: Princeton University Press.Google Scholar
- Barbera, C. Andre. 1980. The persistence of Pythagorean mathematics in ancient musical thought. Ann Arbor: University of North Caroline at Chapel Hill.Google Scholar
- Barbieri, Patrizio. 2004. The speaking trumpet: Developments of Della Porta’s “ear spectacles” (1589–1967). Studi Musicali 33 (1): 205–248.Google Scholar
- Biringuccio, Vanocchio. 1966. Pirotechnia. Trans. and ed. C.S. Smith, and M.T. Grudi. Cambridge, MA: MIT Press.Google Scholar
- Bromberg, Carla, and Ana Maria Alfonso-Goldfarb. 2010. A preliminary study of the origin of music in Cinquecento musical treatises. IRASM 41 (2): 161–183.Google Scholar
- Creese, David. 2010. The monochord in ancient Greek harmonic science. Cambridge: Cambridge University Press.Google Scholar
- Crunelle, Marc. 1991. Acoustic history revisited. http://www.phy.duke.edu/~dtl/89S/restrict/CrunellePaper.pdf. Accessed 9 Mar 2016.
- Gaffurio, Franchino. 1993. The theory of music. Trans. W.K. Kreyszig. New Haven/London: Yale University Press.Google Scholar
- Gozza, Paolo. 2000. Number to sound, The Western Ontario series in philosophy of science. Vol. 64. Dordrecht: Kluwer.Google Scholar
- Masi, Michael. 1983. Boethian number theory: A translation of the De institutione arithmetica. Amsterdam: Rodopi.Google Scholar
- McInerny, Ralph. 1990. Boethius and Aquinas. Washington, DC: The Catholic University of America Press.Google Scholar
- Murdoch, John E. 1963. The medieval language of proportions: Elements of the interaction with Greek foundations and the development of new mathematical techniques. In Scientific change, ed. Alistair C. Crombie, 237–271. New York: Basic Books.Google Scholar
- Rico, Gilles. 2005. Music in the arts faculty of Paris in the thirteenth and early fourteenth centuries. Ph.D. Dissertation, University of Oxford.Google Scholar
- Saito, Fumikazu, and Carla Bromberg. 2015. Measuring the invisible: A process among arithmetic, geometry and music. Circumscribere 16: 17–37.Google Scholar
- Teeuwen, Mariken. 2002. Harmony and the music of the spheres: The ars musica in ninth-century commentaries on Martianus Capella. Leiden: Brill.Google Scholar
- Tronchin, L., I. Durvilli, and V. Tarabusi. 2008. The marvellous sound world in the “Phonurgia Nova” of Athanasius Kircher. Annals of Acoustics’08 Paris, Paris, pp. 4185–4190. http://webistem.com/acoustics2008/acoustics2008/cd1/data/. Accessed 10 Feb 2016.
- Truesdell, Clifford. 1960. The rational mechanics of flexible or elastic bodies, 1638–1788. In Euleri Opera Omnia, 2nd Series, vol. II. Pt. 2, pp. 15–141.Google Scholar
- Walker, Daniel P. 1978. Studies in musical science in the late renaissance. London: The Warburg Institute, University of London/Leiden: E. J. Brill.Google Scholar