CHAPTER IV

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PROBLEM OF DIFFERENT SOUNDS OF THE SAME STRING.

IT had been observed at an early period of acoustical knowledge, that one string might give several sounds. Mersenne and others {37} had noticed[21\8] that when a string vibrates, one which is in unison with it vibrates without being touched. He was also aware that this was true if the second string was an octave or a twelfth below the first. This was observed as a new fact in England in 1674, and communicated to the Royal Society by Wallis.[22\8] But the later observers ascertained further, that the longer string divides itself into two, or into three equal parts, separated by _nodes_, or points of rest; this they proved by hanging bits of paper on different parts of the string. The discovery so modified was again made by Sauveur[23\8] about 1700. The sounds thus produced in one string by the vibration of another, have been termed _Sympathetic Sounds_. Similar sounds are often produced by performers on stringed instruments, by touching the string at one of its aliquot divisions, and are then called the _Acute harmonics_. Such facts were not difficult to explain on Taylor's view of the mechanical condition of the string; but the difficulty was increased when it was noticed that a sounding body could produce these different notes _at the same time_. Mersenne had remarked this, and the fact was more distinctly observed and pursued by Sauveur. The notes thus produced in addition to the genuine note of the string, have been called _Secondary Notes_; those usually heard are, the Octave, the Twelfth, and the Seventeenth above the note itself. To supply a mode of conceiving distinctly, and explaining mechanically, vibrations which should allow of such an effect, was therefore a requisite step in acoustics.

[Note 21\8: _Harm._ lib. iv. Prop. 28 (1636).]

[Note 22\8: _Ph. Tr._ 1677, April.]

[Note 23\8: _A. P._ 1701.]

This task was performed by Daniel Bernoulli in a memoir published in 1755.[24\8] He there stated and proved the Principle of _the coexistence of small vibrations_. It was already established, that a string might vibrate either in a single _swelling_ (if we use this word to express the curve between two nodes which Bernoulli calls a _ventre_), or in two or three or any number of equal swellings with immoveable nodes between. Daniel Bernoulli showed further, that these nodes might be combined, each taking place as if it were the only one. This appears sufficient to explain the coexistence of the harmonic sounds just noticed. D'Alembert, indeed, in the article _Fundamental_ in the French _Encyclopédie_, and Lagrange in his _Dissertation on Sound_ in the _Turin Memoirs_,[25\8] offer several objections to this explanation; and it cannot be denied that the subject has its difficulties; but {38} still these do not deprive Bernoulli of the merit of having pointed out the principle of Coexistent Vibrations, or divest that principle of its value in physical science.

[Note 24\8: _Berlin Mem._ 1753, p. 147.]

[Note 25\8: T. i. pp. 64, 103.]

Daniel Bernoulli's Memoir, of which we speak, was published at a period when the clouds which involve the general analytical treatment of the problem of vibrating strings, were thickening about Euler and D'Alembert, and darkening into a controversial hue; and as Bernoulli ventured to interpose his view, as a solution of these difficulties, which, in a mathematical sense, it is not, we can hardly be surprised that he met with a rebuff. The further prosecution of the different modes of vibration of the same body need not be here considered.

The sounds which are called _Grave Harmonics_, have no analogy with the Acute Harmonics above-mentioned; nor do they belong to this section; for in the case of Grave Harmonics, we have one sound from the co-operation of two strings, instead of several sounds from one string. These harmonics are, in fact, connected with beats, of which we have already spoken; the beats becoming so close as to produce a note of definite musical quality. The discovery of the Grave Harmonics is usually ascribed to Tartini, who mentions them in 1754; but they are first noticed[26\8] in the work of Sorge _On tuning Organs_, 1744. He there expresses this discovery in a query. "Whence comes it, that if we tune a fifth (2 : 3), a _third_ sound is faintly heard, the octave below the lower of the two notes? Nature shows that with 2 : 3, she still requires the unity, to perfect the order 1, 2, 3." The truth is, that these numbers express the frequency of the vibrations, and thus there will be coincidences of the notes 2 and 3, which are of the frequency 1, and consequently give the octave below the sound 2. This is the explanation given by Lagrange,[27\8] and is indeed obvious.

[Note 26\8: Chladni. _Acoust._ p. 254.]

[Note 27\8: _Mem. Tur._ i. p. 104.]

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