Question

Passage:
In the late sixteenth century, the Florentine musical theorist and lutenist Vincenzo Galilei mounted a pioneering experimental challenge to orthodox Pythagorean acoustics. Since antiquity, speculative musical theory had been dominated by the Pythagorean doctrine that musical consonances were governed exclusively by simple whole-number ratios of string lengths. According to classical dogma, the octave corresponded to the linear length ratio 2:1, the fifth to 3:2, and the fourth to 4:3. These numerical proportions were presumed not merely to describe audible phenomena but to manifest an immutable cosmic harmony. Pythagorean scholars maintained that the number four—as the foundation of the sacred tetractys (the integers 1, 2, 3, and 4)—demarcated the absolute boundary of consonant intervals, asserting that physical parameters other than linear length were irrelevant to harmonic ratios.

Galilei, seeking to validate the practical tuning compromises employed by contemporary instrumentalists, conducted systematic empirical tests to determine whether classical ratios applied across diverse physical dimensions. Suspending various weights from gut strings of identical length and thickness, Galilei discovered that producing an octave did not require doubling the weight (a 2:1 ratio), as orthodox theory predicted by uncritical analogy to string length, but rather quadrupling it (a 4:1 ratio). Similarly, producing a musical fifth required a suspended weight ratio of 9:4 rather than 3:2. By systematically extending his investigations to pipes of varying diameters and concave vessels filled with measured volumes of water, Galilei established that harmonic ratios varied according to the specific physical property being manipulated: pitch varied in linear proportion to string length, but according to the square root of string tension and the cube root of resonant vessel volume.

These experimental results led Galilei to repudiate the metaphysical premise that abstract numbers possessed inherent aesthetic power. Instead, he maintained that acoustic consonance was a function of physical mechanics and human sensory appraisal. His demonstrations proved that the number four held no cosmological privilege, given that consonant sounds could be generated by mathematical ratios involving larger integers when tension or volume was altered. Consequently, Galilei’s empirical program not only challenged the authority of scholastic musical theorists, but also helped inaugurate early modern experimental mechanics by insisting that natural phenomena be measured through direct observation rather than deduced from abstract numerological principles.

According to the passage, Vincenzo Galilei's experiments demonstrated that for strings of identical length and thickness, producing an octave required which of the following?