Theory and Practice of Piano Construction / With a Detailed, Practical Method for Tuning

by William Braid White

Now from what we learned of the phenomena of beats, we must conclude that the tempering process when applied to these intervals will generate beats between the sounds that compose each interval. We know that beats must occur when the sounds that form any consonant interval are not quite in tune with one another. We also know that the frequency of the beats depends upon the difference in frequency of the generating sounds. We can, therefore, easily see that those intervals that are subjected to the greatest amount of tempering will produce the greatest number of beats. And further, as the actual frequencies of the sounds increase according to their pitch, it is equally obvious that the tempering will result in greater differences as to actual frequency between the true and the corresponding tempered intervals. Therefore the number of beats that any tempered interval generates varies directly as the pitch of the sounds that form the interval. The higher the pitch, the greater the number of beats. Conversely, the lower the pitch, the smaller the number of beats.

Now we have already noted that the phenomena of beats afford an absolutely precise test for the consonance or otherwise of an interval. If we can estimate the number of beats that should occur between the sounds of any given equally tempered interval, we can always tune such an interval in the Equal Temperament by noting the number of beats and adjusting this to the theoretical number in the calculations. It is not possible accurately to follow the number of beats that are supposed to be between any given intervals in Equal Temperament even when the pitch of the tonic of the interval that is being tuned is precisely similar to the corresponding sound in the calculations. It is not possible, therefore, in practice, to tune with such accuracy as theory would demand, but an approximation may be obtained. If we could secure an absolute standardization of pitch for the pianoforte it would be possible to construct tables that would show the exact number of beats that ought to occur between all the equally tempered sounds within the whole compass. In default of such a method, it is necessary to resort to a variety of tests and to prove the correctness of the tempering of each interval by comparison of the different intervals of various kinds that to which each sound, as it is completed, gives rise. If, for example, we find that any given sound, when tuned, gives the same number of beats with the tenth below as it does with the third below, which is one octave above the tenth, then we have some assurance that the sound in question is properly tempered. If this assurance is confirmed by a complete absence of beats between the given sound and its octave, above or below, then we have an almost absolute assurance as to the correctness of the work.

It is then upon the phenomena of beats that the tuner depends for a guide to the correctness of the work in which he is engaged. By noting the frequency of the beats at some places, or their absence at others, he is able to judge most accurately whether any interval is tuned too sharp or too flat, or whether any octave is tuned purely or the reverse. All good tuning depends entirely upon such estimation of the beats, and the greatest difficulty that the tuner encounters lies in the fact that he must try to equalize the frequencies of the beats between all the intervals of the same kind within the compass of each octave. If this work is well and truly done it properly deserves the name of art, and, indeed, fine tuning is a fine art, one to be acquired by the painful and slow processes of manual practice and mental application. He who overcomes all obstacles to success and masters thoroughly the principles and practice of tuning is an artist in the truest sense.

In applying these principles to the tuning of the pianoforte, the problem that confronts us is to devise a rapid and simple means of tempering each sound within the seven odd octaves of the instrument, and to do this in such a manner that the deviation from purity shall be the same for all similar intervals within the compass of each octave.

Now it follows, from what has gone before, that the tempering of each separate interval, by itself, and without reference to any other, would be a very tedious and inaccurate process. It would, in fact, be quite impracticable to employ such means for intervals that require relatively large deviations from purity, especially in the higher pitched registers. There is, however, a method that largely obviates these difficulties. The middle octave of the instrument, which runs from F below middle C to F above it, is chosen, and the intervals within this octave are so tuned that the thirteen semitones which it contains become equally tempered sounds. The sounds within the next octave above or below are thereupon tuned from the former, each to its octave above or below, and this process is continued until all the sounds upon the key-board have been tuned.

It is easy to see that such a method possesses many and great advantages. All the difficult tempering of intervals that require large deviations from purity is confined to that portion of the piano where beats are most easily estimated; while the rest of the instrument is tuned by means of octave intervals, in which the test of purity is absence of beats, rather than the estimation of any number of them.

The tempering of the intervals in the middle octave is called “laying the bearings” and is the most difficult, as it is the most important, of the various processes incident to the practice of pianoforte tuning. The “accumulation of insensible into almost intolerable errors,” as Mr. Ellis aptly terms it, continually besets the path of the tuner, especially if his preliminary knowledge be imperfect. The true estimation of beats, as generated by various intervals, is an art that is but slowly and painfully acquired, by long practice and training of the ear.

Examination of the pianoforte key-board shows us thirteen sounds within the compass of an octave. In proceeding to the conversion of these into equally tempered sounds, we have more than one method presented to us. We shall, of course, choose the octave which, as stated above, runs from F below middle C to F above it, and shall use, for our purposes, such adjacent sounds as we may consider necessary.

It is usual to take from a tuning-fork the pitch of the sound from which the tuning is begun. These instruments are tuned either to C, or A next above middle C. It is usual, in this country, to tune from C, and we shall, therefore, adopt that method.

Now there are various ways of setting about the “laying of the bearings.” Some tuners work by thirds, others by fourths and fifths; others again use a series or circle of fifths joined by octaves. Whatever intervals are tuned, the idea is to include all the thirteen sounds within the octave and to use, as far as possible, only one or two kinds of intervals.