Determination of the Absolute Pitch of a Note

The vibration of a string stretched between two points depends upon the reflection at either end of the wave motion transmitted along the string. If a succession of waves travel along the string, each wave will in turn be reflected at the one end and travels back along the string and be reflected again at the other end; the motion of any point of the string is, accordingly, the resultant of the motions due to waves travelling in both directions. Premising that a node is a point in the string at which the resultant effect of the incident and reflected waves is to produce no change of position, and that a loop is a point at which the change of position due to the same cause is a maximum, it is evident that if a string is to remain in a state of vibration the two ends of the string which are fixed to the supports must be nodes, and it follows that the modes of vibration of the string must be such that the distance between the two ends contains an exact multiple of half the length of a wave, as transmitted along a uniform string of indefinite length and without obstacles.

It is shown in works on acoustics⁽¹⁾ that a wave of any length travels along such a string with a velocity v where v = sqrt(T/m), T being the stretching force of the string in dynes, and m the mass of a unit of its length expressed in grammes per centimetre.

If τ be the time of vibration of the note, and λ its wave length in centimetres, we have, just as in the case of air,

If n be the vibration frequency of the note

hence

The distance l between the fixed ends of the string being an exact multiple of λ/2, we have

where x is some integer.

Whence

It is this formula whose experimental verification we proceed to describe. The apparatus usually employed for the purpose is known as a monochord or sonometer, and consists of a long wooden box with a wire, fixed at one end and stretched between two bridges by a spring at the other, or by means of a weight hanging down over a pulley. The one bridge is fixed at the fixed end of the string; the other one is movable along a graduated scale, so that the length of the vibrating portion of the string can be read off at pleasure. The measurement of the stretching force T, either by the hanging weight or by the stretching of a spring attached to the end of the box, is rendered difficult in consequence of the friction of the bridge, and therefore requires some care. The pulley itself may be used instead of the bridge if care be taken about the measurement of length. For a fine brass or steel wire a stretching force equivalent to the weight of from 10 to 20 kilogrammes may be employed. This must be expressed in dynes by the multiplication of the number of grammes by 981.

It is convenient to have two strings stretched on the same box, one of which can be simply tuned into unison with the adjustable string at its maximum length by an ordinary tuning-key, and used to give a reference note. The tuning can be done by ear after some practice.. When the strings are accurately tuned to unison, the one vibrating will set the other in strong vibration also; this property may be used as a test of the accuracy of tuning. We shall call the second the auxiliary string.

It is advisable to use metallic strings, as the pitch of the note they give changes less from time to time than is the case with gut strings.

Referring to the formula (1), we see that the note as there defined may be any one of a whole series, since x may have any integral value. We get different notes on putting x equal to i, 2, 3 .... successively.

These notes may in fact all be sounded on the same string at the same time, their vibration numbers being n, 2n, 3n, 4n, .... and their wave-lengths 2l, l, 2l/3, 2l/4... , respectively. The lowest of these is called the fundamental note of the string, and the others harmonics. These may be shown to exist when the string is bowed, by damping the string - touching it lightly with the finger - at suitable points. Thus, to show the existence of the first harmonic whose wave-length is l, bow the string at one quarter of its length from one end, and touch it lightly at the middle point The fundamental note will be stopped, and the octave will be heard, thus agreeing in pitch with the first of the series of harmonics given above.

To obtain the second harmonic bow the string about one-sixth of its length from the end, and touch it lightly with the finger at orle-third of its length. This stops all vibrations which have not a node at one third of the length, and hence the lowest note heard will be the second harmonic, which will be found to be at an interval of a fifth from the first harmonic or of an octave and a fifth from the fundamental tone. We may proceed in this way for any of the series of harmonics, remembering that when the string is damped at any point only those notes will sound that have a node there, and on the other hand, there cannot be a node at the place where the string is bowed; hence the place for bowing and the place for damping must not be in corresponding positions in different similar sections of the wave-curve; if they were in such corresponding positions the damping would suppress the vibration of the string altogether.

The intervals here mentioned may be estimated by ear, or compared with similar intervals sounded on the piano or harmonium.

We shall now confine our attention to the fundamental note of the string. Putting x=1 in formula (1) we get

We have first to verify that the vibration number of the note varies inversely as the length of the string when the tension is constant. This may be done by sliding the movable bridge until the note sounded is at a definite interval from the note of the auxiliary string, with which it was previously in unison. Suppose it to be the octave, then the length of the adjustable string will be found to be one half of its original length; if a fifth, the ratio of its new length to its original length will be 2/3, and so on; in every case the ratio of the present and original lengths of the string will be the inverse ratio of the interval.

In a similar manner we may verify that the vibration frequency varies as the square root of the tension. By loading the scale pan hung from the pulley, until the octave is reached, the load will be found to be increased in the ratio of sqrt(2):1, and when the fifth is obtained the load will be to the original load in the ratio of sqrt(3):sqrt(2).

It yet remains to verify that the vibration frequency varies inversely as the square root of m, the mass per unit of length of the string. For this purpose the string must be taken off and a known length weighed. It must then be replaced by another string of different material or thickness, the weight of a known length of which has also been determined. Compare then the length of the two strings required to give the same note, that is, so that each is in turn in unison with the auxiliary string. It will be found that these lengths are inversely proportional to the square root of the masses per unit of length, and having already proved that the lengths are inversely proportional to the vibration frequencies, we can infer that the vibration frequencies are inversely proportional to the square roots of the masses per unit of length.

We can also use the monochord to determine the pitch of a note, that of a fork for instance. The string has first to be tuned, by adjusting the length, or the tension, until it is in unison with the fork. A little practice will enable the observer to do this, and when unison has been obtained the fork will throw the string into strong vibration when sounded in the. neighbourhood. Care must be taken to make sure that the fork is in unison with the fundamental note and not one of the harmonics. The length of the string can then be measured in centimetres, and the stretching force in dynes, and by marking two points on the wire and weighing an equal length of exactly similar wire, the mass per unit of length can be determined. Then substituting in the formula (2) we get n.

This method of determining the pitch of a fork is not susceptible of very great accuracy in consequence of the variation in the pitch of the note of the string, due to alterations of temperature and other causes.

Experiment. - Verify the laws of vibration of a string with the given wire and determine the pitch of the given fork.

Enter results thus : -

Length of wire sounding in unison with the given fork, 63.5 cm.
Stretching  force (50 Ibs.), 22680 grammes weight = 22680x981 dynes.
Mass of 25 cm. of wire, 0.670 grammes.
Vibration frequency of fork: 256 per sec.

(1)	See Lord Rayleigh's Sound, vol. i. chap. vi.