The practical work of tuning is performed by the aid of certain acoustical phenomena which enable the tuner to distinguish between sounds that are very nearly in unison. As the Equal Temperament requires the slight roughening of all the intervals with the exception of the unison and octave, it is clear that there is great value in a method of estimating, not only the unisonal or non-unisonal condition of two sounds, but also the exact amount of difference that may occur between them. The phenomena mentioned are called “beats,” and it is well that their physical basis should be described here.

From what has been said before, it is clear that musical sounds, generated as they are by the periodic agitation of the air according to fixed laws, are but the audible manifestations of a peculiar form of air-motion. The particular form of air-motion can best be described as a wave. Whenever a sonorous body is excited into vibration it causes the surrounding atmosphere to make motions that correspond to its own. A vibrating body such as we have described (together with the segments thereof) partakes of a motion that may be compared to that of a pendulum. There is a rhythmic swinging back and forth of the body and its segments, with the result that the immediately adjacent layers of air are excited into alternate states of compression and expansion; or, more correctly, of condensation and rarefaction. This rhythmic motion is imparted by the layers of air adjacent to the sonorous body to the next adjacent layers, and so on. The result of this is that a wave is formed, the length of which varies inversely as the number of vibrations performed by the sonorous body. This wave is called a sonorous wave.

Now we know that sounds are at the same pitch when they are generated by sonorous bodies having the same speed of vibration, and it is easy to perceive that, if two such bodies are sounding together, the condensations and rarefactions of the layers of air will synchronize with each other, so that both will be exciting condensations at the same instant and likewise will generate rarefactions simultaneously. And even if the two bodies have not exactly the same speed, the result will be equally simple as long as their speeds bear simple ratios to each other. Thus two bodies which are emitting sounds at the interval of an octave or of a fifth or fourth will generate condensations and rarefactions in such a manner that they will not interfere one with another. But the case is different where two sounds are separated by differences in pitch that cannot be expressed by simple ratios. For example, if one sound be one vibration per second higher than another, it is clear that by the time that the first sounding body has completed its given number of vibrations in one second, the other will be one vibration behind. When, therefore, the vibrations of the first body are continued into the next second the condensation of one wave will be completely synchronous with neither the condensation nor the rarefaction of the other. The obvious result is that at a certain point the condensations of each wave concur while at another point the condensation of one crosses the rarefaction of the other. In the first case we have a considerable augmentation of sound and in the other case a complete silence. As the waves thus approach and recede there is a gradual diminution of sound followed by a complete cessation for a small fraction of a second, and then a gradual increase until the point of greatest augmentation occurs. This latter happens when the two condensations concur, and the gradual rise and fall of the sound correspond to the gradual approach of this concurrence in the first case and to the similar advance of the point of crossing in the second. This phenomenon of alternate augmentation and diminution of sound separated by an almost inappreciable interval of silence occurs whenever two sounds of nearly the same pitch are heard simultaneously. These peculiar changes in the intensity of a sound are denominated “beats.”

This description of the physical nature of “beats” will be sufficient to make clear to us how a recognition of them is of value to the tuner. From what we have just said, it will be observed that the number of beats that may be set up between any two sounds depends upon the difference in the frequency of the two sonorous bodies. So that the number of beats form a true guide to the exact amount of difference between sounds that are nearly in consonance. Thus, if it becomes a matter of tuning a certain interval a little flat or sharp, in order to comply with the requirements of Equal Temperament, the operation may be readily performed by observing the number of beats that are heard between the two sounds when one of them is sharpened or flattened. So that all schemes of tuning must necessarily be founded upon a recognition of this important phenomenon.

At this point it will be well to reiterate the fact that the Equal Temperament owes its popularity and long prevalence to the wonderful facility of modulation which it possesses. While it certainly involves discords and disharmonics that the mesotonic system, for instance, avoided, yet the fact that it does not limit the expression of musical ideas to a few scales, but permits the composer to roam at will through the whole field of tonalities, has given to it a deeply founded popularity that has not yet been seriously challenged. We must bear in mind that the Equal Temperament is the first fact, the “prius,” the “proton hemin,” as it were, of musical performance. Obviously, therefore, the importance of a proper and close adherence to this system in the tuning of fixed-tone instruments cannot be insisted upon too strongly. The reader has already had occasion to examine a comparative table which showed the pitches of a true and of a corresponding tempered scale. He will have noted that the tempered scale errs very greatly in respect to certain intervals. The task of equalizing the thirteen sounds which a fixed-tone instrument allows to the octave, involves in each interval a greater or less divergence from purity, according to the ratio of such interval. Thus we find that the error of a tempered third is greater than that of a fifth, and so on. Now, if the four minor thirds within the compass of an octave be considered, it will be found that the octave to the tonic which is produced from the last of these is a good deal sharper than the octave taken direct from the tonic. Again the octave produced from the building up of the three major thirds within the same compass is very much flatter than the octave taken direct from the tonic. Again, it will be remembered that there are twelve fifths within the compass of seven octaves. The last sound in this progression of fifths is considerably sharper than the sound that is produced by taking a series of seven octaves from the tonic. Without going into figures, we may give the differences thus noted concisely as follows:

In the cases above considered the octaves obtained by building up intervals differ from the straight octave in the following proportions:

Obviously, therefore, it will be necessary to tune all the minor thirds, within an octave, flat by one-fourth each of the ratio given for them. It will also be necessary to tune each of the major thirds sharp by one-third of the ratio proper to those intervals. Likewise each of the perfect fifths must be made flat by one-twelfth of the ratio given above for fifths.

We are thus able to understand just how great divergencies from purity are involved in the Equal Temperament of major thirds, minor thirds and fifths. As far as the other intervals are concerned, it is obvious that if the thirds and fifths are equally tempered and the octaves tuned quite purely, the other intervals will be subjected simultaneously and automatically to a similar process of temper.