112 Chapter 7
From these observations follows an empirical proportionality between the vibrational
frequency of a string and the square root of its tension (here measured by the weight); in
other experiments, he likewise showed that frequency varied inversely as the string ’ s length
and its cross-sectional area, results now called “ Mersenne ’ s laws. ”^19 He also established
similar relations for wind and percussion instruments, demonstrating their general applica-
tion to vibrating bodies. These findings allowed him to carry his result from the ultra-slow
vibrations of his giant string to the realm of more ordinary frequencies. He observes that
a section of the same string about one foot long stretched under an eight-pound weight
sounds in unison with a four-foot organ pipe pitched at the ton de chapelle , one of the
standard pitches in use at the time. From his empirical laws, he deduces that this string
was vibrating at 84 cycles per second, a frequency sufficiently high that, as Galileo had
surmised, it could not have been counted directly.^20 Mersenne goes on, in his methodical
way, to tabulate the frequencies of notes over eight octaves.
He notes that “ a string must beat at least 20 times a second in order to be heard, and
only 42 times a second for its movement to be seen by the eye, nevertheless without being
able to count its returns until it only makes more than ten, ” indicating the greater sensitivity
of the ear to discern these very slow vibrations. Thus, Mersenne ’ s experimental technique
essentially depends on the ear even as it explores realms of frequency that are no longer
aurally discernible.
At the same time, Mersenne became interested in aspects of sound that would not
depend on the observations of a trained ear. By applying the results of his empirical laws,
he was able to show that “ a deaf man can tune a lute, viol, spinet, and other string instru-
ments and find the sounds he wishes, if he knows the length and size of the strings. ” He
provides a “ harmonic tablature for the deaf ” that enables them to find the visible charac-
teristics of different notes they might be asked to produce ( figure 7.4 ). Perhaps this was
addressed to his friend Descartes, who by 1638 described himself as “ almost deaf. ”^21 In
the following generation, Joseph Sauveur made important contributions to acoustics (even
providing that name to the field) though profoundly deaf and mute until age seven.^22
Conversely, Mersenne demonstrated that “ one can know the size and length of strings
without measuring or seeing them, through the means of sounds, ” so that hearing can
substitute for the other senses.^23
A similar blend of practical musical consideration and theoretical speculation character-
izes Mersenne ’ s other investigative initiatives. He may have been the first to measure the
speed of sound and to show its independence of pitch and loudness, a proposition he tested
by various kinds of echoes. Mersenne ’ s experiments involve using language itself to probe
the speed with which the echo is formed; “ it is certain that all sorts of echo that repeat
seven syllables pronounced in the time of a second must cover the distance of 485 feet, ”
which he compares to the firing range of an arquebus.^24 He repeated the syllables Bene-
dicam Dominum ( “ Let me bless the Lord ” ) at higher and lower pitches, softly and loudly,
in foggy and clear weather, to determine that the speed of their sound, measured by the