In the case of the fifth, it will be observed that the difference in frequency between the fundamentals is 128, while this number also expresses the smallest difference which occurs between any of the particles.

In the fourth, the smallest difference in frequency is 83. There is apparently, a difference of 2 between the seventh overtone of the funda mental and the fifth overtone of the higher note, but this is because for simplicity we have taken the frequency of this note as 341, when it ought to be 341.3.

In the major sixth, the smallest difference is 84.
In the major third, the smallest difference is 84.
In the minor third, the smallest difference is 50.

Now of the intervals considered above the most consonant is the fifth, and the consonance decreases as we pass to the minor third. This decrease in the consonance is accompanied by a decrease in the smallest difference in frequency of the partials of the two notes, so that in the case of the minor third we are approaching the limit (47 beats per second) at which discord begins.

Next let us take a case where, although the difference between the frequencies of the fundamentals is greater than in several of the cases above, yet the consonance is not so good, and see whether we can account for this dissonance by the production of beats between the partials. For this purpose we may take the notes g and g➡, and a slightly mistuned fifth.

In the case of c and g, the difference between the fundamentals is 113, and so these tones will not produce discord. The second overtone of c and the first overtone of g, however, differ by 30, and are therefore dissonant, and it is to the beats produced by these that the dissonance of the interval is due. In the case of c and g, the second overtone of and the first overtone of g differ in frequency by 32, while the sixth overtone of c and the fourth overtone of g differ by 48, and the dissonance of the interval is thus accounted for. In the untrue fifth there will be eight beats per second between the second overtone of the lower note and the first overtone of the higher, sixteen per second between the

fifth overtone of the lower and the third overtone of the higher, and twenty-four between the eighth overtone of the lower and the fifth of the higher. Hence it will be seen why it is that an untrue fifth is dissonant, and how the ear is able to detect want of correctness in such an interval. We may in the same manner examine the other intervals, but this task is left to the reader.

Since, when an interval is untrue, those partials of the two notes which, if the interval were true, ought to be in unison, will be in a condition for producing beats, it follows that the greater the number of common partials, and the stronger these partials are, the greater will be the discord produced by mistuning the interval, and so the greater the accuracy with which the ear can adjust such an interval. In the case of the perfectly consonant interval, the octave, all the partials of the higher note are in unison with partials of the lower. In the fifth, the alternate overtones (1, 3, 5, &c.) of the higher note are in unison with partials of the lower. In the fourth, every second overtone (2, 5, &c.) of the higher are in unison with partials of the lower. In the major sixth, major third, and minor third, one overtone of the higher is in each case in unison with one partial of the lower, but as the consonance decreases, it is a higher and higher, and therefore less important partial that is in unison. Thus an interval is more consonant the greater the number, and more important, that is, the lower, the partials which are common to the two notes. When, instead of compound tones, simple tones are employed, the above explanation does not account for the fact that while a true octave or fifth is consonant, an untrue, that is, slightly mistuned, octave or fifth is dissonant. It is to be noted that with pure tones the accuracy with which the ear is able to detect an untrue interval is very considerably less than with compound tones, so that the explanation given above is to this extent supported. Helmholtz has explained the dissonance of simple tones as being due to the beats produced by combination tones. Thus, suppose we have an untrue octave, the frequency of the tones being 256 and 515. The number of beats is 259, and therefore cannot produce the discord which is heard. These two tones will produce a difference tone, of which the frequency is 515-256 or 259, and this tone will give three beats per second with the tone 256. It is to these beats that Helmholtz attributes our power of distinguishing the untrue interval. In the same way, suppose we have an untrue fifth, the frequencies being 256 and 380. The first difference tone has a frequency of 380 - 256 or 124, and will not produce a discord with either primary. A secondary difference tone will be produced between the first difference tone and the lower primary of frequency 256 - 124 or 132, and this secondary difference tone will produce eight beats per second with the first difference tone, hence the discord.

319. Timbre. The quality or timbre of the notes given by different instruments is produced by the overtones which accompany the fundamental. In general those notes in which the fundamental is relatively

strong, the overtones being few and feeble, are of a mellow character. On the other hand, notes in which the overtones are numerous and strong are harsher, and have a so-called metallic sound.

In dealing with the effects of the overtones on the timbre of a note, there are three points to be considered: (1) the relative frequencies of the particles, (2) their relative intensities, and (3) the relations that exist between the phases of the partials.

That the first two of these conditions will have an important bearing on timbre is evident from a consideration of the curves given in Fig. 273,

(a) and (c), which represent the resultant curves obtained by the combination of sine curves which are all harmonics of the same curve, but in which the harmonics present are different. In the case of a note in which the corresponding partials are present, we can easily see that the timbre may be very different in the two cases.

FIG. 274.

(From Ganot's "Physics.")

In order to determine what partials were present in any given note, Helmholtz used a series of resonators, each of the form shown in Fig. 274. The open end a of the resonator is turned towards the source of sound, while the other and smaller opening, b, is connected to the ear by means of an india-rubber tube. If the given note contains a partial which corresponds in pitch to the natural pitch of the resonator, then this partial will cause the resonator to "speak."

Koenig has devised the form of resonator shown in Fig. 275, in which the volume of the enclosed air can be altered, and thus the pitch of the note to which the resonator responds also altered.

Instead of using the ear to detect whether a resonator responds to a given note, use may be made of a manometric flame, the capsule of which is connected with the inside of the resonator. On this principle Koenig has constructed the instrument shown in Fig. 276. A series of resonators are tuned so as to respond to a given tone and to its harmonics, each resonator being connected by means of a tube with a manometric capsule. When a note, of which the fundamental agrees in pitch with the pitch of the lowest resonator, is sounded near this instrument all the resonators which agree in pitch with the partials that are present will respond, and the corresponding flames will be affected, which effect can be observed by looking at the flames in a rotating mirror. When an open organ - pipe, of which the fundamental corresponds to the lowest resonator, is sounded gently the fundamental is the only resonator that responds. If, however, the pipe is sounded more strongly, the resonators corresponding to the first five harmonics respond, the response of the third harmonic being stronger than that of the second. If a closed pipe of the same pitch is sounded strongly, then the even harmonics are all absent, as we should expect from the discussion given in § 306. The third harmonic is fairly strong, while the fifth is only feeble.

FIG. 275

In the case of the note given by a bowed violin-string, the first seven harmonics are present, and it is owing to the presence of this large number of partials that the violin owes the piercing character of its notes. In the case of the piano, the partials 1, 2, 3 are fairly strong, while the partials 4, 5, 6 are more feeble, while the position at which the strings are struck is so chosen that the seventh partial is absent. The reason for this is that the seventh partial, when present, forms a dissonance with the sixth and eighth.

The frequency of the sixth, seventh, and eighth partials being 6n, 7n, and 8n, where n is the frequency of the fundamental, the interval between the 6th and 7th is 6/7, and that between the 7th and 8th is 7/8. Now neither of these intervals is consonant. If, however, the 7th partial is absent, the interval between the 6th and 8th is 3/4 (a fourth), and this is consonant. Hence when the 7th partial is wanting, all the partials up to and including the 8th are consonant. Although the 8th and 9th are dissonant, yet since the loudness of a partial decreases rapidly with the order of the partial, the dissonance produced by the 6th and the higher partials is practically negligable.

The partials of an organ-pipe have been investigated by Raps in

another way. Two rays of light are caused to form interference bands (§ 378), and while one ray passes altogether through the external air, the other passes through the air situated at the node of an organ-pipe. The alternate condensations and rarefactions cause the density of the air to alter, and hence the velocity of light passing through the air varies in the same way. The result is that when the pipe is sounded the interference

bands vibrate backwards and forwards in the same period as the changes of density of the air in the pipe. If then the bands are received on a rotating drum covered with photographic paper, a wavy line will be produced, and from the character of this line the nature of the vibrations of the air in the pipe can be seen. A series of traces obtained in this way from an open pipe blown with gradually increasing wind pressure is