To prove the incorrectness of Helmholtz's statement that beats do not colesce into musical sounds, but that the ear will distinguish them as a rumbling noise, even when their number rises as high as 132 vibrations per second, Rudolph Koenig has constructed a series of tuning forks, recently presented by President Morton to the Stevens Institute of Technology. The following table exhibits the number of vibrations per second of these forks, the ratios of their vibrations when two are sounded together, the number of beats produced, and the resultant sound:

Vibrations per second.Ratio.Beats.Sounds.
3904: "61:6496Sol
3936: "123:12880Mi
3968: "31:3264Ut
3976: "497:51260Si
3989.3: "187:19253.3La
4000: "125:12848Sol
4010.7: "47:4842.7Fa
4016: "251:25640Mi
4024: "503:51236Re
8064: "63:6464Ut
8096: "253:25648Sol
8106.7: "95:9642.7Fa
8112: "507:51240Mi
8120: "1015:102436Re
8128: "127:12832Ut

On sounding two forks nearly in unison, the sound heard corresponds to a number of vibrations equal to the difference of the numbers of vibrations of the forks.

On sounding two forks, one of which is nearly the octave of the other, the ear perceives a sound, which is that given by vibrations whose number equals the difference in the number of vibrations of the higher fork and the upper octave of the lower fork.

Koenig has also found out the laws of the resultant sounds produced by other intervals than the octave, and has extended his researces to intervals differing by any number of vibrations, as may be seen from the above table.

His conclusion is that beats and resultant sounds are one and the same phenomenon.

Thus, for example, the lowest number of vibrations capable of producing a musical sound is 32 per second; in like manner, a clear musical sound is produced by two simple notes of sufficient intensity which produce 32 beats per second.

Koenig also made a very ingenious modification of the siren for the purpose of enabling Seebeck to sound simultaneously notes whose vibrations had any given ratio. It is furnished for this purpose with eight disks, each of which contains a given number of circles of holes arranged at different angular distances. A description of this instrument, which is also the property of the Stevens Institute, and of Seebeck's experiments is thus given in a letter by Koenig himself.


Effects produced when the isochronism of the shocks is not perfect.


In order to produce a note, the succession of shocks must not deviate much from isochronism.

If the isochronism is but little impaired, we obtain a note corresponding to the mean interval of the shocks.

If the intervals between the shocks are alternately t and t', and if the difference between t and t' is slight, we obtain the two notes t+t' and (t+t')/2. If the intervals between the shocks are alternately t, t', and t", we obtain the two notes t+t'+t" and (t+t'+t")/3.

Disk No. 1 has--

 On circle No. 1 12 holes, angular distances t=30°

" " 2 24 " " " 15°

" " 3 36 " " " 10°

" " 4 36 " at irregular distances.

" " 5 36 " distances t= 10½°, t'=l0°,t"=9½°

" " 6 36 " " 11° 10° 9°

" " 7 36 " " 16° 14°

" " 8 36 " " 16½° 13½° 

Circle No. 8 produces the two notes of circles 1 and 2; circle No. 7 the same, but the low note is stronger than in 8.

Circle 6 produces the notes of circles 1 and 3, and so does circle 5, but in the latter the low note is stronger than in 6.

Circle 4 produces a noise approximating only to the note of circle 3.

By pulling out one of the buttons of the wind chest, we admit the air through eleven holes at a time, having an angular distance of 30° and directing it against the corresponding circle of holes on the turning disk. If the arrangement of holes is not repeated identically twelve times on the same circle, we cannot, of course, make use of the above arrangements of holes of the wind tube, and we must then employ one of the movable brass tubes, which communicate with the interior of the wind chest by means of rubber tubes and stopcocks. The experiment with disk 1, circle 4, for example, requires the use of one of these two tubes, while the perforated wind tube of the wind chest may be used with all the other circles of the same disk.


If t is much less than t', while t' is a multiple of t, the note (t+t')/2 disappears, and the notes t+t' and t are heard.

Disk No. 2 has--

 On circle No. 1 12 holes, distances 30°

" " 2 36 " " 10°

" " 3 48 " " 7½°

" " 4 60 " " 6°

" " 5 24 " " t= 5°, t'=25°

" " 6 24 " 6° 24°

" " 7 24 " 7½° 22½°

" " 8 24 " 10° 20° 

Circle 8 produces the notes of circles 1 and 2; circle 7, those of 1 and 3; circle 6, those of 1 and 4; and circle 5, the note of circle 1 and of its sixth harmonic.


If the same circular arc is divided into m and n equal parts; that is to say, if mt=nt', we obtain the notes m and n.

Disk No. 3 has--


On circle No. 1 24 holes, distances 15°

" " 2 24 " " 15° & 27 holes, 13-1/3°

" " 3 24 " " 15° " 30 " 12°

" " 4 24 " " 15° " 32 " 11-1/4°

" " 5 24 " " 15° " 36 " 10°

" " 6 24 " " 15° " 40 " 9°

" " 7 24 " " 15° " 45 " 8°

" " 8 24 " " 15° " 30, 36, & 48 holes 

Circle 1 produces a single note, circle 2 a second, circle 3 a third, circle 4 a fourth, 5 a fifth, 6 a sixth, 7 a seventh, and 8 a perfect chord.


Experiments to prove that the shocks may proceed from two or several different places to conspire in the formation of a note, provided that the isochronism of the shocks is sufficiently exact, and that the shocks are produced in the same direction.

Disk No. 4 has--

 On circle 1 24 holes.

" " 2 36 "

" " 3 23 "

" " 4 12 at an angular distance of 10° from the holes

of circle 3.

" " 5 12 holes at an ang. dist. of 20° from those of circle 3

" " 6 12 " " " 0° "

" " 7 12 " " " 15° "

" " 8 12 " " " 15° " 

1. If from the same side two currents of air at an angular distance of 15° are directed against circle No. 8 of 12 holes, we obtain the octave of the note produced by the same circle if only one current is used.

The wind-chest is provided with a special arrangement for this experiment. By pulling out button 8, we give vent to 12 currents of air spaced like the twelve holes of the disk; on pulling out button 9 we also produce 12 currents, but they are situated just between the first. Each of these two buttons pulled out alone will produce the same note corresponding to 12 holes, but drawn together they produce the octave, or the note of circle 1.

2. If two currents of air are directed against two similar circles whose holes are situated on the same radii, we obtain the same result.

In this experiment, circles 7 and 8 are sounded by pulling out buttons 7 and 9.

3. When two currents of air are directed on the same radius against two circles of similar holes arranged alternately, these circles sounded simultaneously will produce the octave of the note which one of them would give alone.

This experiment is performed by sounding circles 6 and 7 and pulling out buttons 6 and 7.

4. If we direct three currents of air on the same radius against three similar circles having holes alternating by a third of the distance between two holes of the same circle, the three circles together produce the fifth of the octave (Note 3) of a single circle.

Circles 3, 4, and 5 sounded together emit the note of circle 2.

(By sounding only two circles, 3 and 4, or 4 and 5, we make the same experiment with two circles as disk No. 2 enabled us to make with circle 8 alone; also, by sounding circle 3 alone, we obtain the note corresponding to 12 holes; then pulling out button 4, the notes corresponding to 12 and 36 holes are heard suddenly and very strongly; but as soon as circle 5 is sounded also, the note of 12 disappears completely, and we have left only that corresponding to 36 holes.)


Effects of interference produced by shocks in opposite directions.

1. If we direct against a circle of holes two currents of air in opposite directions, the note obtained with a single current is very much weakened, if the two currents reach the holes simultaneously. If the impulses are not isochronous, the intensity of the note is increased.

2. If the two currents are directed against two circles of the same number of holes, the effect is the same as for the two preceding cases.

3. If two currents of air are directed against two circles, one of which has twice as many holes as the other, we obtain only the low note if every shock of one is isochronous with every shock of the other.

We obtain the notes of both circles, one of which is the octave of the other, if there is no isochronism between the shocks.

Disk No. 5 has three circles of 36, 36, and 72 holes. The air currents are directed against the circles of holes through the movable tubes, made so that they can be detached at pleasure. All these experiments require great precision in the arrangement of these wind tubes. To make sure that the tubes are simultaneously before two holes of the disk, it is well to put little rods through the holes, reaching into the wind tubes, and to remove them only when the tubes are firmly attached. The experimenter should be careful also to place the two tubes exactly at the same distance from the turning disk. It is clear that notwithstanding all these precautions we never obtain perfect interference, but only the weakening of notes that ought to disappear entirely if all the arrangements were made with mathematical exactness, and also if the ear could have absolutely the same position with regard to impulses produced in opposite directions.



Disk No. 6 has--

8 circles of holes to the number of 1, 2, 23, 24, 25, 47, 48, 49.

Circles 3 and 4, 4 and 5, 6 and 7, and 7 and 8 ought to produce as many beats as circle 1 produces simple shocks; and circles 3 and 5, 6 and 8, as many beats as circle 2 produces simple shocks; but we must content ourselves in these experiments with a much less perfect result, for the following reasons: The disk never being rigorously plane, alternately approaches the single wind pipe and recedes from it. No matter how slight this deviation is, every sound given by a single circle is heard with periodical intensities which complicate the phenomenon. This inconvenience could be avoided by placing several wind-pipes around the circle; but while we can extend the period of the holes in two circles (whose difference is 1) around the whole circle by blowing through a single wind tube, we would be compelled to limit it to the distance between two wind tubes, and it would become too short; for, when the disk rotates with a velocity sufficient to produce notes high enough and intense enough, the beats become too numerous to be easily perceived.

Besides these provisions, which sufficiently illustrate the points to which we desire to call especial attention, Koenig also furnishes two more disks.

The seventh contains 8 circles having 48, 54, 60, 64, 72, 80, 90, and 96 holes respectively. The 1st, 3d, 5th, and 8th will produce a perfect chord when the air is admitted through the 11 holes in the wind chest; with one wind tube the entire gamut may be obtained.

Finally the eighth disk contains 8 circles of holes, whose numbers are in the ratio of 1:2:3:4, etc., and which may be used to illustrate harmonics. C. F. K.