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Thomas B.  Greenslade, Jr.

Kenyon College

Gambier, Ohio 43022 USA


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To a physicist, a tonometer is a calibrated secondary frequency standard. The best tonometers were the sets of tuning forks made by Rudolph Koenig. He brought them to the 1876 Philadelphia Centennial Exposition, and a set was bought by the United States government and assigned to the United States Military Academy at West point. Other tonometers were sets of steel rods suspended from their nodal points. The least accurate, but certainly the least expensive, were the harmonica-like sets of vibrating reeds from Max Kohl. The use of the clock fork and the vibration microscope for comparing standard tuning forks is discussed.


 Overtone Apparatus

The interesting-looking device in Fig. 1 recently came into my collection.  It rather resembles a harmonica on steroids, but a look at the 1925 Max Kohl catalogue (ref. 1) shows that it was actually an “Overtone Apparatus, consisting of nine reed pipes with wind chest and wind regulator, for the 1st to 9th overtone of c-1 = 64 Hertz (ut1 = 256 Hertz).”  The bellows are in the upper portion of the device, and the reeds are put into play by pulling out the stop knobs across the front, lifting the upper portion of the apparatus by several cm and letting it drop under its own weight.  The sound, quite naturally, resembles that of an old-fashioned reed organ, and pleasing chords can be formed with various combinations of frequencies.

Ton F1(a)ed

Fig. 1 Small Overtone Apparatus open

Fig. 1   Small Overtone Apparatus in the Greenslade Collection


Just below it on the catalogue page is a larger version, with the first thirty two overtones of c-1 = 64 Hertz, this time running up to ut6 = 2048 Hertz.  Then I realized that I had photographed this apparatus many years earlier at Duke University, where it had been in regular use for a course for non-science majors.  It is shown in Fig. 2.  Again, there is a bellows in the top, formed, as usual, from sheepskin.  The black plastic electricians tape is a later addition!

Ton F2(a)

Fig. 2 Large Overtone Apparatus at Duke University

Fig. 2   Large Overtone Apparatus at Duke University



And now to the word Tonometer.  To a physicist this is a set of secondary frequency standards; to the ophthalmologist, it is a device for measuring the internal pressure of the eyeball, useful for diagnosing glaucoma.  I am interested in the first definition, for at the top of the Kohl catalogue page are a series of five tonometers, all resembling larger versions of the device in Fig. 2.  The two largest ones have 129 reeds each, covering the range from 256 Hz to 512 Hz and then from 512 Hz to 1024 Hz, both in steps of “about” two Hertz.  The use of the qualifying “about” rather surprises me, unless it is to warn the buyer that this is not a primary standard.

Vibrating harmonica-type reeds are thin, elongated pieces of brass, placed a small distance above a parallel rectangular orifice through which the wind passes.  They may be actuated by air from below or from above; in the Kohl design the bellows is above.  The physics of the vibrating reed is governed by a wave equation that involves the fourth derivative of the transverse displacement with respect to the position along the bar, and the second derivative with respect to time.  The resulting solutions, due to Lord Rayleigh, are anharmonic, and the reeds, if blown gently, resonate at the fundamental.  This frequency depends on the length of the reed and the reciprocal of the square of its length.  Tuning a reed can be accomplished by reducing the thickness of the metal at the clamped end to lower the frequency, or by filing the tip to increase it; this is done with a fine jeweler’s file.

This immediately leads to the question of the standards on which the vibrating reed tonometer is based.  Fortunately, I have seen examples of the tuning fork tonometer sets brought by Rudolph Koenig (1832-1901) to the 1876 Centennial Exhibition in Philadelphia as part of a large display of his instruments.  Despite the efforts of Joseph Henry and others, not enough money was raised to buy the entire collection.  However, in 1882 $10,000 was paid for a portion of the collection, which was presented to the United States Military Academy at West Point, New York.  This included a tuning fork tonometer, with the forks mounted on resonating bases.  The frequencies of these forks were quite accurate, for they were surely adjusted by Koenig himself.  Only two short runs remain, and are shown in Figs. 3 and Fig. 4.  The forks in Fig. 3 run from 484 Hz to 496 in steps of 4 Hz, although the bases are marked with the usual half-period markings and read 968 to 992.  Another very large set, although not mounted on resonating boxes, is at the National Museum of American History at the Smithsonian Institution in Washington, D.C.

Fig. 3 Four tuning forks from a Koenig Tonometer

Fig. 3   Four tuning forks on resonating bases from the large Koenig Tonometer, now at the United States Military Academy. Each one is separated by 4 Hz

Fig. 4 Three tuning forks of a large Koenig Tonometer

Fig. 4   Three tuning forks on resonating bases from the large Koenig Tonometer, now at the United States Military Academy


One can infer that the Kohl factory owned a set of these graduated tuning forks and used them to tune the vibrating reed tonometers that they sold.  But we can follow the standardization back another step by looking at the instrument in Fig. 5.  This was the vibration microscope that allows the frequency of a tuning fork to be compared with a standard one that has almost the same frequency.  The apparatus in the figure is at Amherst College in Massachusetts, and is listed at 140 francs in the 1889 Koenig catalogue (Cat.  No.  234i).

Fig. 5 Koenig's Vibration Microscope

Fig. 5   The Vibration Microscope, as made by Koenig


The basis of the operation lies in the use of Lissajous Figures to compare two frequencies. (Ref. 3)  Since readers of eRittenhouse may not be familiar with this technique, it is illustrated in Fig.  6.  Two tuning forks are mounted at right angles to each other and have small mirrors mounted on their facing tines.  A light beam strikes one fork and is then reflected by the other.  The light beam is caused to oscillate up and down by the first reflection and side to side by the second.  If the two forks have the same frequency, the beam of light draws a circle, an ellipse or a straight line.  If the ratio of the two frequencies is two to one, the beam draws out a figure-eight on the screen.  This technique was invented by Joules Antoine Lissajous (1822-1880) in 1857, and follows on the early analysis by Nathaniel Bowditch in 1815.


Fig. 6 Two tuning forks, oscillating at right angles to each, produce a 2-to-1 Lissajous figure.

Fig. 6    Two tuning forks, oscillating at right angles to each, produce a two-to-one Lissajous figure.   From A. Ganot, Elementary Treatise on Physics, 11th ed., trans. and ed. by E. Atkinson (Longmans, Green, and Co., London, 1883) p. 239

The original vibration microscope in Fig. 6 was invented by Lissajous, and the form in which the reference tuning fork is driven electromagnetically is due to Hermann von Helmholtz (1821-1894).  The eyepiece of the microscope is fixed to the apparatus, but its objective is attached to one prong of the tuning fork.  Thus, when the reference tuning fork is driven electrically, the field of view vibrates up and down.  The microscope is focused on a point on the tuning fork under test that is vibrating from side to side.  This point thus vibrates up and down and side to side, thus tracing out a Lissajous figure.  If the two forks are slightly different in frequency the figures will not be stationary, but as the frequency of the fork under test is adjusted a standard figure is produced.

To determine absolute frequencies, Koenig designed and sold a basic instrument, the clock fork (Fig. 7), the description of which he published in 1880.  This was a large tuning fork oscillating at a frequency very close to 64 Hz.   The vibratory motion of the fork controlled the rate of a mechanical clock, in much the same way as the pendulum and escapement mechanism was used to control the rate of the standard pendulum clock of the day.  Like the pendulum clock, energy was fed back by the escapement mechanism to keep the tuning fork vibrating continuously.  The frequency of the fork was adjusted (by moving small masses up or down the massive tines) until the clock kept essentially perfect time.  Under these conditions, the frequency of the fork was 64 Hz to a precision of 10-4 Hz.  Koenig found that the frequency of the steel forks was a function of temperature, going down by a factor of about 0.01% for a temperature rise of one Celsius degree.  Consequently, the clock fork was equipped with a sensitive thermometer to allow temperature corrections to be made.

Fig. 7 Koenig’s Clock Fork (1900)

Fig. 7   Koenig’s Clock Fork, from J.A. Zahm, Sound and Music, 2nd ed.
(A.C. McClurg Co., 1900)


For his own use, Koenig made a tonometer which consisted of 154 forks ranging in frequency from 16 Hz to 21845.3 Hz.  He used either the method of beats or Lissajous figures to inter-compare the frequencies of the series of tuning forks.  His standard frequency was 256 Hz, four times that of the 64 Hz clock fork.

Another form of Tonometer made by Koenig is shown in Fig. 8.  It is at Amherst College, and cost 80 Francs in the 1889 Koenig catalogue.  Here a series of steel rods, suspended by cords from points 22.4% in from either end, are struck in the middle, sending them into transverse vibrations.  The governing differential equation is the same as that for the vibrating reed, but here the boundary conditions require that each end be free.  The frequencies are anharmonic, and by striking the rod gently only the fundamental is excited.  The frequency is dependent on the inverse square of the length of the bar.  Once the lower frequency rods are tuned, usually by using beats with a standard source, such as a tuning fork whose frequency is derived from that of the clock fork, the frequencies of the higher rods can be determined just by measuring their lengths.  This allows an intriguing possibility: the 32,768 Hz sound produced by the shortest rod in the tonometer can be heard only by bats.

Fig. 8 Resonating Rod Tonometer by Koenig,

Fig. 8   Resonating Rod Tonometer by Koenig, at Amherst College


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Ref. 1   Price List No. 100, Volume II, Physical Apparatus for the Mechanics of Solids, Liquids and Gases, the Wave Theory, Acoustics and Optics (Max Kohl A.G., Chemnitz (Germany), ca.  1925), p. 373.

Ref. 2    Alexander Wood, Acoustics (Dover Publications, New York, 1966), pp 420-424.

Ref. 3    Thomas B.  Greenslade, Jr., “All About Lissajous Figures”, Phys. Teach. 31 (1993) 364-370.


[Editor’s note added 2016/06/19:  Readers are reminded of the following recent book on Keonig and his contributions to the understanding of sound in the 19th c.

David Pantalony. Altered Sensations: Rudolph Koenig’s Acoustical Workshop in Nineteenth‐Century Paris. (Archimedes: New Studies in the History of Science and Technology, 24.) xxxvi + 372 pp., illus., tables, bibls., index. Dordrecht: Springer, 2009.]