The meaning of the discussion is, therefore, that the fifths within the octave where the “bearings” are being laid must each be tuned flat by two beats. Or, rather, that the higher sound of every fifth must be flatter by two beats than if it were in consonance with the lower sound.
Again, if we tune the F below middle C two beats sharp of the latter (which is equivalent to tuning the fixed pitch sound C two beats flat of F) we shall obtain a properly tempered fifth. Now, if the octave above this F be taken it will be found to form a sharp fourth with middle C. For example, if the pitch of middle C be 264, then the pitch of F below it, in pure intonation, is two-thirds or 176. Assuming that the F be then tempered so as to be sharp by two vibrations, it will have a frequency of 178. The octave to this is 356 But this latter is a fourth above C 264 and should, therefore, be 352 Consequently we see that the fourths in Equal Temperament are to be tuned sharp ascending, or conversely, flat descending. As already explained, the beat-rate in the “bearings” should be nearly one per second or three beats, while the sound of the interval remains audible.
When the deviations from purity are as slight as in the cases that we have been considering, it is by no means easy to determine, at all times, whether the note that is being tuned is sharp or flat of its tonic. For the beats occur similarly in either case and few ears can determine the relative sharping or flatting without some extraneous aid. Fortunately, however, we have a variety of tests open to us, which for completeness and accuracy leave nothing to be desired.
To take a concrete example, during the “laying of the bearings” we first tune the F below middle C, then the G below middle C, and then the D above G. When we reach this last note, we find that a sixth has been obtained; namely, F—D. Now if the notes already tuned have been tempered, so as to be too flat, the resultant sixth will beat too slowly, and, conversely, if the tuned notes be too sharp the sixth will beat too fast.
This test may be amplified when we proceed to the next interval, D—A. When this latter note has been tuned, we have the triad F—A—C. F—A is a major third, and by referring to previous calculation we see that as such it must, when properly tempered, be considerably sharp. By noting the beats of the major third and likewise the beats of the sixth we may correct the tuning of all the sounds with which we have hitherto dealt. The same process is, of course, carried on throughout the whole process of “laying the bearings.” The major thirds and sixths are tested continually as the tuning proceeds, and thus is provided a sure guide to the correctness of the fourths and fifths.
The correct beat-rates for the major thirds may be stated as about eight per second, while that for the major sixths is approximately eleven in the same period. Of course, as already stated, these rates per second cannot be measured with accuracy, but with practice one soon discovers by ear the proper roughness in each case, and is thus enabled to estimate the beat rates without much trouble. Every opportunity of examining the work of good tuners should be taken by the observer, who should note carefully the beat-rates which they assign to each kind of interval. In this way he will provide himself with practical examples of tempering of intervals which will be of great value to him.
Having thus determined the proper beat-rates for each of the intervals that are used in the “laying of the bearings,” we may proceed to the further consideration of that convenient method for tuning the middle octave that has already been demonstrated.
In order to facilitate comprehension of the argument, the following table is given, showing graphically the sounds that are tuned in laying the bearings and the tests and trial chords—
The white notes are those to be tuned. The black notes are those already tuned.
The kind of deviation from purity of interval and the beat-rates are as follows: