fraction can thus be determined for each interval, in the manner exemplified in the case of the Fifth. We will call this fraction the vibration-fraction of the interval in question. The accompanying table gives, in the second column, the vibration-fractions corresponding to the intervals named in the first; and, in the third, states the consonant or dissonant character of each interval.

It is noticeable that the vibration-fractions of the dissonant intervals involve higher numbers than those of the consonant intervals do; the latter, with

the solitary exception of the Minor Sixth, having no number exceeding 6, while the former bring in 9, 15 and 16.

36. By the help of this table we can calculate the vibration-numbers of all the notes within a single Octave which belong to the Major or Minor keys as soon as the vibration-number of their tonic is given. For instance, let middle C of the pianoforte (vib.-no. 264) be the tonic. From the fourth line of the table we see that the vibration-number of D must be to 264 in the ratio of 9 to 8. It must

The student should work out the remaining cases for himself.

The complete results for the Major scale are as follows:

In order to extend the scale an Octave upwards we have only to multiply each vibration-number by 2. A second multiplication by 2 will raise it by another Octave, and so on. Conversely, in order to

pass to the Octave below we divide each vibrationnumber by 2. To descend a second Octave we repeat the operation, and so on.

Thus the pitch of the tonic absolutely fixes the pitch of every note in the scale of which it is the starting-point.

Before proceeding to investigate the mechanical equivalent of the third element [§ 24] of a musical sound, its quality, we will briefly examine, in the next chapter, a subject possessing an important bearing on that enquiry.

CHAPTER III.

ON RESONANCE.

37. WHEN a sounding body causes another body to emit sound, we have an instance of a very remarkable phenomenon called resonance. The German term for it, 'co-vibration' (Mitschwingung), possesses the merit of at once indicating its essential meaning, namely the setting up of vibrations in an instrument, not by a blow or other immediate action upon it, but indirectly as the result of the vibrations of another instrument. In order to produce the effect, we have only to press down very gently one of the keys of a pianoforte, so as to raise the damper without making any sound, and then sing the corresponding note loudly into the instrument. When the voice ceases, the instrument will continue to sustain the note, which will then gradually fade away. If the key is allowed to rise again before the sound is extinct, it will abruptly cease. A similar experiment may be tried, as follows, on any pianoforte which allows the wires to be uncovered. Each note is, as is well known, produced by two or by three wires. Having, as in

the previous case, raised one of the dampers without striking the note, twitch one of the corresponding wires sharply with the finger-nail, and then wait a few seconds. The vibrations will, during this interval, have communicated themselves to the other string or strings belonging to the note pressed down: if, therefore, the first wire be now stopped by applying the tip of the finger to the point where it was at first twitched, the note heard will continue to be sustained by the remaining wire or wires.

A more instructive method of studying resonance is to take two unison tuning-forks, strike one of them, and hold it a short distance from the other. The second fork will then commence sounding by resonance, and will continue to produce its note if the first fork be brought to silence. It is essential to the success of this experiment that the two forks should be rigorously in unison. If the pitch of one of them be lowered by causing a small pellet of wax to adhere to the end of one of its prongs, the effect of resonance will no longer be produced, even though the alteration of pitch be too small to be recognised by the ear. Further, the phenomenon generally requires a certain length of time to develope itself; for, if the silent fork be only momentarily exposed to the influence of its sounding fellow, hardly any vibration is communicated. The reso