Abstract
A multitude of stringed instruments make use of plucking as a source of excitation. These range from the harp of Egyptian antiquity to the oriental Koto. Some feel that the most elegant of all plucked instruments is really the harpsichord. (See later illustrations.)
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Notes
- 1.
See Appendix B for a derivation of the wave equation and various solutions to the vibrating string problem. Note that the wave equation itself is an approximation valid for very small amplitudes compared to the length of the string. The drawings given here exaggerate the size of the deflection for the sake of illustration. Typically, the deflection is only a few millimeters in a string 4-ft long.
- 2.
As discussed in Chap. 1, a “hard” reflection produces a change in sign of the running-wave amplitude.
- 3.
Using the trigonometric identities \(\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B\), Bernoulli’s solution may also be written as a sum of oppositely directed running waves:
$$\displaystyle \begin{aligned} y(x,t) = \sum_{n=1}^\infty A_n \left[ \frac{1}{2} \sin \left(\frac{n \pi x}{L} + 2\pi n F_0 t\right) + \frac{1}{2} \sin \left(\frac{n \pi x}{L} - 2\pi n F_0 t\right) \right]\,.\end{aligned} $$(3.4) - 4.
It is also a simple matter to integrate the expression in Eq. (3.4) numerically using the sine portion of the Fourier analysis program in Appendix C.
- 5.
Some upright pianos may be exceptions. The geometry of an upright piano (and possibly of a square piano) could make striking the string at the midpoint feasible. With a concert grand, it would be out of the question since the hammer shanks would have to be about 4-ft long in the extreme bass.
- 6.
In addition, we have neglected the inharmonic character of vibrating strings, a subject that we will come back to in the discussion of pianos. That property alters the periodic nature of the vibration, but it is less important in harpsichords where the strings are relatively thin.
- 7.
The function f n(t) increases linearly with harmonic number, n, and can be obtained by taking the time derivative of Eq. (3.2):
$$\displaystyle \begin{aligned} f_n (t) = -2\pi nF_0 \sin{}(2\pi nF_0t )\,. \end{aligned} $$The further simplification in Eq. (3.7) occurs by use of the trigonometry identity
$$\displaystyle \begin{aligned} \sin{}( A \pm B ) = \sin A \cos B \pm \cos A \sin B \,.\end{aligned} $$ - 8.
That is, nulls occur that are independent of the plucking point for values of n such that
$$\displaystyle \begin{aligned} \frac {n\pi a}{L} = m\pi\ \text{and}\ \frac{n\pi x_0}{L} = \frac{2m-1}{2}\pi\ \text{for}\ m =1, 2,3\ldots\end{aligned} $$ - 9.
In general, the magnetic field (H), which is normal to the coil, will vary laterally over the string motion. The field can be expanded as a power series in the lateral coordinate (y), giving
$$\displaystyle \begin{aligned} H(y) =H_0 + H_1 y+ H2 y^2 + \cdots \end{aligned} $$where y = 0 corresponds to the rest position of the string. The term involving y 2 will produce second harmonics of a sinusoidally oscillating string whose velocity varies as sin 2πF 0t. The latter follows from the trigonometry identity,
$$\displaystyle \begin{aligned} \cos 2A = 2 \cos^2 A - 1\,. \end{aligned} $$(3.8) - 10.
Each stop is described in terms of the length of an open pipe required to produce the same pitch as the lowest C on the keyboard.
- 11.
Anyone who has ever tuned a harpsichord can imagine what a nightmare the 1710 instrument by J. A. Hass would produce with two sets of strings at 16-, 8-, 4-, and 2-ft pitch! (See Hubbard 1965, Plate XXVII.)
- 12.
Modes of vibrating membranes are treated in Appendix B. The stiffness in the soundboard plays a role analogous to the tension in a string, where for the one-dimensional case the wave velocity is \(c = \sqrt {T/\mu }\), T is the tension, and μ is the mass density per unit length. Stiffness varies as the 3rd power of the thickness.
- 13.
Not all sounds produced by a harpsichord originate with the soundboard. The mechanical motion of the jacks can create distracting noises. Once, when harpsichordist and composer Joyce Mekeel and I were recording a piece she had written called “Textures” for harpsichord and clarinet, she complained about strange noises in the recording. Finally, we put the microphone directly behind her head and the “strange” sounds disappeared. The music desk had been shielding her ears from the jack noises.
- 14.
Hubbard (1965, p. 327) suggests that thicker gut stringing was probably used on some Italian instruments. But, he was “bound to report that gut strings on Italian instruments sound very badly.” He also noted that Bach had written for the Lauten Werck, or lute harpsichord, an instrument using gut strings.
- 15.
- 16.
The “beating” process described here is a linear one that arises simply by adding two waves at different frequencies. Consider two equal amplitude sine waves whose frequencies are proportional to A and B. Adding the two trigonometry identities,
$$\displaystyle \begin{aligned} \sin{}( x \pm y) = \sin x \cos y \pm \cos x \sin y\,, \end{aligned} $$and letting x = (A + B)∕2 and y = (A − B)∕2, one gets the result,
$$\displaystyle \begin{aligned} \sin A+ \sin B = 2 \cos \left[ \frac{A-B}{2}\right]\sin\left[ \frac{A+B}{2}\right]\,. \end{aligned} $$Thus, when A ≈ B, the resultant wave is at the average of the two frequencies and is amplitude-modulated at half the difference frequency—an effect called “beating.” One tunes the strings by adjusting the beat frequency, (A − B)∕2, to zero.
- 17.
Mussorgsky was emulating the hexachordal mode of Russian church music given in the Obikhod, which may have forced him into that large number of flats. He actually did slip into E-major once in his lifetime in the “Prelude” to Khovanshchina, but that was probably forced upon him by the minor third present in the church bell on Red Square signaling matins. (Mussorgsky’s church bells usually tolled in C♯.)
- 18.
Curiously, the Well-Tempered Scale appears to have been a Chinese invention. Early approximations have been traced to Ho Che’eng-t’ien in the fifth century. But, the first use of a scale in which the intervals were based on the 12th root of two has been credited to Chu Tsai-yü in the sixteenth century (Yung 1980, p. 261).
- 19.
When the two-manual 1770 Taskin harpsichord first came into the Yale University Collection in 1957, there was a rectangular hole in the underside of the instrument large enough to permit inserting one’s head. One heard quite a glorious sea of sound in there when the instrument was played.
- 20.
This technique was extended to pianos by an interior decorator living in Princeton during the 1940s. She covered her entire Steinway inside and out with the same blue wallpaper used in the rest of the living room, not to mention the closet and entrance doors. Once inside the room, one might not find the way out, let alone locate the piano! Hubbard commented (1965, p. 22), “The Ruckers harpsichords are charming, but their naïve crudity is to the sophisticated Italian harpsichord as a cuckoo clock to Brunelleschi’s Duomo.” (One Ruckers motto was AVDI VIDE ET TACE, which might be freely translated as “Listen, look, and be quiet.”)
- 21.
For example, see Vermeer: “The Music Lesson” (1662–1664), “The Concert” (1665–1666), “Lady Standing at the Virginal” (1672–1673), and “Lady Seated at the Virginal” (1675). Jan Steen’s painting entitled “The Music Master” (1660) almost certainly shows a Ruckers instrument, but Steen signed his own name on the harpsichord. Also see Gerrit Dou, “Woman at the Clavichord” (1665).
- 22.
A later restoration by Frank Rutkowski and Robert Robinette (done in March 1987) using 0.023″ yellow and red brass (90% copper and 10% zinc) on the low C was made after the present data were taken.
- 23.
Hubbard himself established a company (in a barn in northeastern Massachusetts) that sold precut parts as kits. Kits are also available from the Zuckerman company and various premade parts are sold by the B & G Instrument Workshop in Ashland, OR. However, “do-it-yourselfers” should be warned that it takes about 300 expert-person hours to build a two-manual Taskin and that you have to complete five or six before you start turning out good ones. Back in the 1970s, I asked a Hubbard associate what he could do for a kit that was poorly completed by one of my children. (At that time, the two-manual “Taskin” kit sold for about $2400.) The reply was, “We take them up to the second floor of our barn and toss them out the hay-loft window.”
- 24.
A stereo recording of the Hubbard–Rephann instrument made by the author of several Bach pieces played by harpsichordist Lola Odiaga is available on CBS Discos. (See Odiaga 1974.)
- 25.
Oboist Robert Bloom once told me that he thought “they all sounded like old bed springs.” On the other hand, Sir Thomas Beecham felt that the sound was more “like two skeletons copulating on a corrugated tin roof” (Atkins and Newman 1978, p. 34.) But, Beecham was probably referring to one of the later English models which Ripin et al., p. 234 say sounded like loud brass bands compared to the woodwind-like quality of the French instruments.
- 26.
Yale chemist Martin Saunders (private communication) proposed a possible solution to the tuning problem: A computer-controlled electric-power source feeds current through each string separately at about 1/2 Watt maximum per string. Increasing the current heats the string, lowering its pitch (and vice versa), providing required tuning changes within a 1- or 2-s thermal response time. A computer-controlled FFT continuously samples the output from a microphone above the strings and is used to monitor the pitch of each note continuously. The computer then controls the current through each string to keep the entire instrument in tune. Saunders suggested that one could also have the program provide continuous harmonic tuning as the music changed keys. But, it is not clear how the method could distinguish between two harmonics on different strings that were of the same frequency. (The method might just result in a puddle of molten strings on the soundboard.)
- 27.
Private communication.
- 28.
Amazingly, Ralph Kirkpatrick once did a recording of Bach’s Well-Tempered Clavier on the clavichord without using any vibrato at all.
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Bennett Jr., W.R. (2018). Plucked Strings. In: Morrison, A. (eds) The Science of Musical Sound. Springer, Cham. https://doi.org/10.1007/978-3-319-92796-1_3
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