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: