CHAPTER XIV.

TUNING AND TONE REGULATION OF THE PIANOFORTE.

The art of tuning the pianoforte is one of considerable complexity and obscurity. During all the time that has elapsed since the key-board instrument first came into being, controversies innumerable have raged over the multifarious questions that the practice of the art implies. This distressing state of affairs is primarily due to the fact that a system of “tempering” the sounds produced by key-board instruments is necessary, in order that playing in more than one key may be possible.

The whole matter of musical intonation was treated with some completeness in the early part of this work, and the reader may be expected to comprehend the principles of the Equal Temperament upon which the tuning of the pianoforte and of all other fixed tone instruments is now universally based. It is not within our province, in the course of a treatise upon the principles of pianoforte construction, to venture too deeply into the quagmires that surround the aspirant for the honors of the tuner. The discussion of musical intonation and the Equal Temperament was made for the principal purpose of acquainting the student with the reason for the peculiar construction of scales, as to their string lengths, and to make clear the _raison d’etre_ of the frequent divergences from theoretical proportions of strings and tones that we have been obliged to note.

The scale designer is better equipped for his task, however, if he possess a working knowledge of the principles upon which the science and art of tuning are based. This is the justification for the space and time devoted to the exhibition of certain of these principles in the earlier portion of this book. As a further contribution to this useful body of knowledge, we shall point out here the general scheme whereby the tuner proceeds to the execution of his important, indeed essential, portion of the whole work.

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:

The octave produced from minor thirds is sharper in the ratio 1296:1250.

The octave produced from major thirds is flatter in the ratio 125:128.

The octave produced from fifths is sharper in the ratio 531441:524288.

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.

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.

Of all these methods, the shortest, easiest and most accurate is that which employs fourths and fifths only. It is used in such a manner that, by tuning a circle of fifths and fourths, the last sound tuned provides the octave to the first, thus completing the circle and the octave of tempered sounds.

In this method we proceed as follows:

1. Pitch C is tuned by the tuning fork. 2. F below pitch C is tuned, being a tempered fifth. 3. G below pitch C is tuned, being a tempered fourth. 4. D above G is tuned, being a tempered fifth. 5. A below D is tuned, being a tempered fourth. 6. E above A is tuned, being a tempered fifth. 7. B below E is tuned, being a tempered fourth. 8. F sharp below B is tuned, being a tempered fourth. 9. C sharp above F sharp is tuned, being a tempered fifth. 10. G sharp below C sharp is tuned, being a tempered fourth. 11. D sharp above G sharp is tuned, being a tempered fifth. 12. A sharp below D sharp is tuned, being a tempered fourth. 13. F above A sharp is tuned, being a tempered fourth.

This last F is the octave to the first F tuned, and should coincide exactly with the latter.

The reader will, of course, realize that the tempering of these various intervals must be tested by means of the generated beats. Helmholtz, in “Die Lehre der Tonempfindungen,” calculates that the tempered fifths should average .6 of a beat per second at standard pitch within the octave that we are treating. This is equivalent to three beats in five seconds. But it is impracticable to measure the generated beats upon the pianoforte in this manner. The tone of the instrument is too evanescent and fleeting. We may, however, attain to a very fair approximation. If, for example, each fifth be tuned so that two distinct beats are heard before the sound dies away, it will be found that the beat-rate is a near approximation to the calculated average. The two beats that we speak of occur in about three seconds, while the Helmholtz rate is three and one-third seconds for two beats.

Again Helmholtz gives an average beat-rate for tempered fourths in the same octave; namely, one per second. If we tune the fourths so that we hear three distinct beats we shall likewise obtain a very fair approximation to the calculated beat-rate.

We showed above that the last sound produced by the building up of a progression of twelve fifths is sharper than the sound produced by the piling up of seven octaves from the same tonic. The two sounds thus produced ought to coincide, for the compass of twelve fifths and of seven octaves is the same. We concluded, therefore, that the Equal Temperament required the flattening of all the fifths.

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:

Upper notes of fifths must be tuned flat so as to give two distinct beats.

Lower notes of fourths must be tuned flat so as to give three distinct beats.

Upper notes of major thirds must be sharp so as to give approximately eight beats per second.

Upper notes of major sixths must be sharp so as to give approximately eleven beats per second.

If the laying of the bearings has been accurately performed, the chords that are produced by the combination of the various sounds will be found to have very nearly the same roughness and to generate approximately the same number of beats. The best work is that which most closely approaches this standard.

After the bearings have thus been laid, it is necessary to continue the tuning of the instrument above and below the octave already treated. Now, since the octave to any given sound has exactly double the frequency of the former, and since octaves are to be tuned purely, it follows that we need only tune the remainder of the instrument by octaves up and down from the “bearings.” The consonance of an octave is determined by the absence of beats, and, consequently, we have only to follow this rule to reproduce, in all parts of the piano, deviations from purity _relatively_ the same as those which we have been at such pains to secure.

Of course, the _actual_ number of beats in any given interval varies as the frequencies, and, consequently, the frequency of the beats increases regularly in the higher registers, while it similarly decreases in the lower portions. Thus the deviations from purity become much greater in the higher portions of the compass and incorrect laying of the bearings is, therefore, productive of more and more disagreeable results as the scale ascends. On the contrary, in the lower register these inaccuracies are less productive of irritation, as the frequencies of the beats become continually less.

Such, then, is the process of tuning the pianoforte in Equal Temperament. We have made no attempt to go into those practical details that are concerned with the ear-training, the manipulation of the hammer, or other cognate matters. These are entirely “_ex provincia_” of this work.

Nevertheless the person who digests the foregoing statement of the laws and methods of the art will be well equipped to pass upon the correctness of tuning, and this is sufficient for our purpose.

The work of tone-regulation opens up varying but not entirely dissimilar fields of research. The material of which the pianoforte hammer is constructed has an important influence upon the coloring of the sound that it draws from the string. In order that the influence of the pianoforte hammer in tone-coloring may be understood clearly, we shall investigate the matter with some completeness.

We have already investigated the compound nature of the sounds excited by the pianoforte strings. We know that the nature of these sounds varies as the method of excitement and as the nature of the resonance apparatus that surrounds the strings. Now the pianoforte strings are excited by being struck, and we have already noted that the point of striking must be carefully chosen. Further, we know that the amount of metal framing, the manner of adjusting the bridges, the nature of the sound-board and innumerable other details must be taken into consideration.

But by the time that the pianoforte comes into the hands of the tone-regulator, it is out of his power to affect the construction in any fundamental points. He is able to change only two things. These are the striking point and the condition of the hammer-heads. Even the former cannot be changed to any great extent. In the highest treble, however, it often becomes necessary to bend the hammer-shanks slightly in order that a more correct striking point may be obtained. But this must be done with great discretion and caution.

The object of tone-regulation is to ensure an agreeable tone and perfect evenness of quality throughout.

Obviously, this is so intimately bound up with the whole construction that it may be said that the tone-regulation begins with the drawing of the scale and is never finished until the pianoforte itself is completed. This would be a perfectly proper statement, but we have here to consider the final touches, as it were, that the tone-regulator may give to the tonal equipment.

When the instrument comes into his hands the hammers are still covered with the hard outer skin of the felt head. This must be removed with a sand-paper file. It then is necessary to sound each tone slowly and carefully, first loudly, and then less so. It will be found that some of the hammers produce a harsh and disagreeable timbre, while others may be too soft and mushy. Under all circumstances it is far better to endow the instrument, as far as is possible, with a quality that shall be comparatively mellow and round. Brilliancy cannot be forced artificially without spoiling the whole quality and imparting a thinness and roughness that is most disagreeable.

If the hammers are too hard at the crown, they have the acoustical effect upon the strings of exciting the upper dissonant partials to undue prominence. This occurs from the fact that when the head is hard it rebounds instantly from the string, and thus does not damp any of the dissonant partials. On the other hand, if the head be soft, the felt clings for a fraction of a second longer to the strings and effects the damping to a greater or less degree, according to the relative softness or hardness of the head.

It is thus possible, by discreet manipulation of the felt, to influence the character of the sounds to no small degree. The method is to pick up the felt, when it is to be softened, with a set of felt needles mounted in a handle for the purpose. It is essential to note whether the sound is the same when the hammer strikes with great force as it is when it strikes gently. If, for example, a gentle pressure on the key gives an agreeable quality, but harder strokes on the key destroy this, then we see that the crown of the hammer head is soft enough, but the felt cushion underneath is too hard. Fine needles are, therefore, employed to dig into the lower cushion of felt, while disturbing the consistency of the upper crown as little as possible. On the other hand, when the quality of sound is hard under all conditions, the upper surface of the hammer-head must be treated by picking it with heavier needles. Hardening of the felt may also be undertaken by covering the hammer-head with a damp cloth and then applying a hot iron.

The whole work is primarily one of practice and experience. No directions can do more than give an outline of the processes and the physical reasons for them. It is well, however, to lay down the laws that underly these processes, in order that practice may be supplemented and improved with theory.