(3) Determination by the Monochord.-One of the earliest successful attempts at accurate determination of pitch was made by Perronet Thompson. For this end he revised and perfected the ancient instrument of Euclid and Pythagoras, the monochord. According to his construction it was five feet long, ten inches broad, and six deep; the wire was of steel the twentieth of an inch in diameter, containing 145 feet to the pound avoirdupois, breaking with a weight of 300 lbs. The load required to produce tenor C of the pianoforte was from 240 to 250 lbs. The sound was brought out by the application of a well-rosined bow, and had the strength of a violoncello. The method of using the above apparatus for the enharmonic tuning of an organ, will be described in a later chapter. Here it will be sufficient to note the direct physical method of measurement with such an instrument. A string is tuned to a given note, and its vibrations are determined by knowing the stretching weight, the weight of the wire as stretched, and the vibrating length of the string. The following is the formula usually adopted, as given by Mr. Ellis in his excellent communication to the Society of Arts.

1

Let V =

second.

Pitch, or number of double vibrations in one

W = Number of grains in the stretching weight.

S = Number of grains in one inch length of stretched

string.

L =

Number of inches in vibrating string.

Hence SL =

Weight of vibrating string; which, cut off, weighed and measured, gives L, SL, and S.

P being the length of the seconds pendulum = 39·14 at Greenwich, and the constant 3.14159.

The string is brought into sensible unison with the given note by shortening or lengthening the wire, and cut to the correct length. It is carefully measured for L, and weighed for SL. The weight with its attachments is weighed for W.

In this way Dr. Smith, in the year 1755, in the month of September, tuned a wire to give a note two octaves below the D pipe of the organ in Trinity College, Cambridge; arriving

1 Journal of Society of Arts, May 25, 1877, "On the Measurement and Settlement of Musical Pitch," by Alexander J. Ellis, F.R. S., &c.

at the result of D = 262 or mean tone C

a whole tone below the usual pitch at the present time. The above method, in spite of its theoretical beauty, is so liable to constructive difficulties that it is of little or no practical value for the determination of pitch.

A somewhat better form of the monochord for this purpose was introduced by Griesbach, and is preserved in the collection at South Kensington. It consists of a thick gutstring stretched over a body like that of a double bass. It was tuned two octaves below the note to be measured. Then a fine point being attached to one part of the string, a long strip of paper was passed over it at uniform velocity, and in passing was pricked by the point at every double vibration of the string. The notes being then counted and multiplied by four, the pitch of the fork was approximately determined. The employment of this method, also open to numerous sources of error, rendered the fork issued by the Society of Arts too sharp by 37 of a semitone.

The monochord, although it produced good results in the hands of Perronet Thompson, for tuning correctly the different notes of the scale, is hardly so satisfactory as a means of determining absolute pitch. Scheibler sums up his long experience with it thus :-"Had it been possible to obtain exact results with a monochord, I could not but have succeeded, during the many years that I devoted to it, in tuning the forks of my scale correctly. My ear, and those of all others, were satisfied with the purity of the notes on instruments tuned by my monochord forks. But my mind would not be satisfied, because my results were not constant. When for example, one monochord showed me that a certain fork was one stroke of the pendulum too sharp, another monochord gave it as too flat. I became convinced that a mathematical monochord could not be constructed. I had also discovered that the string could not be protected from the warmth radiated by the observer's body, even when it was so thoroughly covered that there was only just space enough left for striking it. The string of a monochord, from this cause, does not remain for 30 seconds at the same pitch, but varies constantly by one-tenth to one-half of a double vibration."

In another place he estimates the possible error of the monochord at five double vibrations.

(4) Graphic Methods have the advantage of substituting a purely mechanical operation for a process requiring the assistance of an accurate musical ear. In their simplest form,

they may be typified by attaching a small point or style to the prong of a tuning-fork, and allowing this to trace its movements upon a piece of smoked paper or glass allowed to travel

steadily before it. If the fork be not sounding, the point will describe a straight line. But if it be first set in vibration, the attached point will constantly move backwards

and forwards, and the wave-line remains as a permanent image of the motion performed by the fork during its musical oscillations. In practice it is best to wrap the paper round a rotating cylinder; rotatory movements being, as a rule, more easy to regulate, and steadier than those in a straight line. The cylinder may, moreover, be made to move in a spiral, instead of performing a simple axial rotation, and thus the line may be indefinitely extended within a limited space.

The curve obtained from any simple harmonic motion is one of those denominated curves of Sines; and it may, as Helmholtz remarks, be made to reproduce the motion of the vibrating points, by cutting a narrow vertical slit in a piece of paper, and placing it over the curve-tracing: if this latter be drawn uniformly under the slit from right to left, the point, seen through the slit, will appear to move backwards and forwards precisely in the same manner as the original tracer attached to the fork.

This method is susceptible of very considerable accuracy: and indeed has been employed for the exact determination of both its factors. Chronographs, in which a steadily vibrating tuning-fork is the standard, tracing its oscillations on a sheet of blackened paper carried past it, have been constructed: a fine example was exhibited at the Loan Exhibition of Scientific Apparatus. A second style, beside that on the fork, is usually made to note, by an abrupt motion on the same strip of paper, the exact instant of any astronomical or other phenomenon, such as the flight of a projectile, which it is desired to determine. On the other hand, the Phonautograph of Scott and the Vibroscope of Duhamel, which is practically the apparatus described above, aim at measuring the pendular vibrations of a sounding body in terms of the cylinder's rotation.

(5) The Phonautograph.-This instrument replaces the tuning-fork shown above, by a hollow barrel about 18 inches long and a foot in diameter. One end is open, the other is closed except by a tube carrying on it a stretched membrane. Upon the membrane is fixed a bristle, which moves with the membrane, and acts as a style. In order that this should not be at a node, the membrane is touched by a moveable piece, which, being made to touch the membrane first at one point and then at another, enables the observer to alter the arrangement of the nodal points. The bristle is thus made to coincide with a loop, that is a point at which the vibrations of the membrane are at a maximuın.

When a sound is produced, the air in the cavity of the barrel and the membrane vibrate in unison with it, and the style is made to trace on a rotating blackened surface the

Fig. 39.-Combination of two parallel vibratory movements.

form of these vibrations. Each undulation corresponds to a double vibration of the style. The fork used in the vibroscope is retained, and traces beside the tracing of the style