The history of natural philosophy is also a history of music. Music defined Pythagorean philosophy and played an integral role in the Scholastic quadrivium. Both Galileo and Huygens played the lute and had prominent composer fathers. The list of seventeenth-century natural philosophers that addressed musical problems—Kepler, Galileo, Descartes, Beeckman, Mersenne, Huygens, and Newton, among others—reads like a roll call of great names in early modern science. These natural philosophers sought, above all, to solve a problem that went all the way back to the Pythagoreans, namely the problem of consonance: why do some sounds seem consonant and pleasing while others seem painfully dissonant and jarring? However, in stark contrast to the spectacular new theories in physics, astronomy, and anatomy emerging at the time, music during the Scientific Revolution gradually fell out of the scientific discourse. The problem of consonance was never satisfactorily solved and an entirely new science of sound, known to us as acoustics, took its place. Music thus presents an unusual and illustrative historical case: a branch of natural philosophy that did not become a science.
(By convention, the classical science of music is often referred to as “harmonics,” although to scholars both ancient and early modern it was simply musica. )
Whereas the modern word “music” serves as a catchall term for songs, symphonies, hymns, etc., the Latin musica before the seventeenth century bore no necessary connection to audible sound. Medieval scholastics, foremost among them the philosopher Boethius (c. 480–524), recognized three principle types of musica: musica mundana was seen as the music of the world and heavens; musica humana dealt with relations between people, ethics, and the body; and (last and usually least) musica instrumentalis, which stood for audible music either sung or produced by instruments. Thus the music of the quadrivium was the mathematical study of time, ratios, and proportions that purportedly governed both the celestial and human realms. Musical sound was merely the sensed expression of these inherent harmonies in nature and musica instrumentalis was man’s attempt at “aping his creator” as Kepler eloquently put it. The skill of writing and performing music was exactly that: a skill of artisans and craftsmen learned through constant, menial practice rather than the inborn genius philosophers cherished.
Kepler’s planetary melodies for the six known planets plus
the moon, as presented in the Harmonices mundi of 1619
(Incidentally, this is what Kepler’s cosmic harmony would actually sound like. Listen at your peril.)
The Scientific Revolution saw music transformed from an abstract notion of harmony in nature to the concrete study of physical sound. The first great revolution in music came from Vincenzo Galilei, composer, lutanist, and (most famously) father of Galileo. Scholastic thought maintained that harmony resulted directly from specific proportions in sonorous bodies. For example, an octave occurred when two strings, one half as long as the other, vibrated simultaneously. The consonance resulted from the harmonious 2:1 proportion and vibration was correlated with, but not the cause of, the music. Humans enjoyed consonant sound for the same reason that a rational mind favored simple, precise proportions. Vincenzo, however, noted that solid bodies such as hammers (as in the original myth of Pythagoras) did not follow the same proportions as vibrating strings: for two hammers to produce an octave, their weight had to be in a 4:1 proportion. Vincenzo himself did not propose much of a solution to this conundrum but his work turned music theory on its head by rejecting an almost two-thousand-year-old tradition. As the only natural philosopher of our story whose main occupation was as a composer and performer of musica instrumentalis, Vincenzo shifted attention away from harmonious proportions and towards the physical construction of the harmonious object. The stage was now set for a physical study of music that emphasized not abstract numerology but concrete matter and its properties.
Galileo Galilei one-upped his father. In Two New Sciences Galileo—after recounting experiments performed with a vibrating chisel and glasses filled with water—made the first connection between rate of vibration of a body and the tone heard. Harmonious ratios (2:1 for an octave, etc.) resulted not from string length but from relative frequency; an octave was made when a string (or bell, hammer, etc.) vibrated twice as fast as another string. Following the rise of mechanico-corpuscular views of nature, philosophers such as Isaac Beeckman were just beginning to understand sound as the transmission of pressure (either as waves or as rectilinear pulsations) from bodies to the “drum of the ear” as he called it. Galileo’s discovery fit the mechanical worldview perfectly: two notes were consonant, according to Galileo, if the sound pulses the emitted coincided often, an idea known to historians as the coincidence theory of consonance.  An octave’s pulses would coincide every other instance and thus sound very consonant. Thus the work of the Galileis neatly exemplify that a key trend in early modern science—replacing ancient, abstract theory with physico-mathematical explanations backed by experiment—applied to music as much as any other domain of natural philosophy.
With the Scholastic theory of music brought under scrutiny, natural philosophers of the seventeenth century set out to determine a new physics of music. In his Compendium musicae (1618) Descartes became the first to explicitly declare that “the object of music is sound,” a statement not at all self-evident at the time. Marin Mersenne, the first to identify a mathematical relation between the length, weight, tension, and frequency of a vibrating string (now known as Mersenne’s law), also became nearly the last natural philosopher to believe in an abstract harmony of nature. Mersenne insisted that nature would only produce harmonies and in his studies of overtones (the partial tones that, as we now know, make up the musical harmonic series) he adamantly denied hearing dissonant harmonics. Mersenne, in short, studied sound only insofar as it informed his understanding of harmony. He asked why certain sounds were harmonious and what role harmony played in nature, but sound in general was not the main interest for Mersenne or his contemporaries. Music, not sound, remained primary.
By the end of the seventeenth century the natural philosophy of music had largely been replaced by the fledgling science of sound soon to be known as acoustics. In 1696 the French mechanic and early acoustician Joseph Sauveur modified Descartes’ definition of music, adding that “the object of music is sound in the sense that it is agreeable to the parts of the ear.” Moreover in 1701 Sauveur defined acoustics as the study of all sound and therefore “a science superior to music.” Among Sauveur’s key insights was that harmonics need not be harmonious in the traditional sense of “pleasing to the ear.” The seventh harmonic (forming a highly dissonant ratio of 7:1 with the fundamental) is the best example of this; Sauveur broke with Mersenne and recognized the seventh harmonic as a legitimate part of acoustics despite it never being used in music. Whereas previous natural philosophers like Mersenne had emphasized harmony and studied sound only insofar as it explained the underlying harmony of nature, Sauveur asserted that sound—not just harmony, but all sound, regardless of how consonant or dissonant it might seem to fallible human ears—was now the fundamental phenomenon under scientific investigation.
Jean-Philippe Rameau, seen here
looking extremely French
A very different science of music emerged in the Enlightenment by the hand of Baroque composer and theorist Jean-Philippe Rameau. In his Traité de l'harmonie of 1722 Rameau famously declared that “music is a science which should have definite rules; these rules should be drawn from an evident principle; and this principle cannot really be known to us without the aid of mathematics.” He thus earned the title of “the Newton of harmony.”  Rameau certainly recognized the tremendous social, cultural, and intellectual prestige surrounding the new science and sought to secure for music a place at the scientific table. Starting with the principle of the major triad—itself derived from the intervals of the harmonic series—Rameau derived all consonant intervals as inversions of the notes in a given triad. Rameau’s Traité de l'harmonie became the gold standard of musical education and many of his principles are still in common use today. However Rameau rarely, if ever, directly dealt with acoustics and did not supply physical justifications for his principles of harmony, stating outright that “we shall leave the task of defining sound to physics.” The end result was a science of music that resembled other sciences in method—a well-formed, empirico-analytic system of study and practice—but not in subject matter. It was not physics, mechanics, geometry, or natural philosophy. A science of music perhaps, but formulated by a musician for musicians with no effort to relate its principles to the rest of the sciences.
(A few of Rameau’s harpsichord pieces can be heard here. They are composed with the kind of mathematical precision that one might expect from the “Newton of harmony.”)
The history of the science of music is, in some sense, a classic Scientific Revolution narrative: out with Scholastic ideas of numerology and abstract qualities, in with the mathematical study of physical bodies and their sonorous properties. Away with God and His divine harmony; the new science of acoustics concerned only the nature of sound, never the harmony of nature. Above all the belief that nature somehow preferred harmony, that certain sounds were intrinsically consonant and others were intrinsically dissonant, lost scientific credibility. Science could not differentiate between musical sound and sound in general—only human beings could do that. This marked the fundamental distinction between acoustics and music theory, a distinction embodied by the acoustician Sauveur, who could not play music to save his life, and the composer Rameau, who did not even read Sauveur’s acoustical work when formulating his Traité de l'harmonie. Hence by the midst of the Baroque era science and music had very little overlap. The word “music” came to denote primarily the audible art form, for all other conceptions of musica lost meaning. In an Enlightenment world that demanded precise quantification and mechanization in science, acoustics thrived but the old natural philosophy of harmonics simply had no place.
 Thomas Kuhn employs the term “harmonics,” whereas H. Floris Cohen simply calls it “music.” Thomas S. Kuhn, “Mathematical vs. Experimental Traditions in the Development of Physical Science,” The Journal of Interdisciplinary History 7, no. 1 (July 1, 1976): 1–31; H. F. Cohen, Quantifying Music : The Science of Music at the First Stage of the Scientific Revolution, 1580-1650 (Dordrecht Netherlands ; Boston: D. Reidel Publishing Company, 1984).
 Cohen, Quantifying Music, 32.
 John Hawkins, A General History of the Science and Practice of Music (London : Printed for T. Payne and Son, 1776), Vol. 2, p. 901.