Piano Tuning and the Allied Arts
-- William Braid White

TABLE OF CONTENTS

Prefatory Note

Chapter I. Mechanics of the Musical Scale

Chapter II. On the Vibration of a Piano String

Chapter III. Temperament

Chapter IV. Practical Tuning in Equal Temperament

Chapter V. Mechanical Technique of Tuning

Chapter VI. The Modern Piano

Chapter VII. Soundboard and Strings

Chapter VIII. The Action and Its Regulation

Chapter IX. The Hammer and Its Relation to Tone

Chapter X. Repair of the Piano

Chapter XI. Elementary Pneumatics

Chapter XII. General Construction of Player Mechanisms

Chapter XIII. Repair of Player Mechanisms

BACK TO PIANO TITLE PAGE

TO

THE CONFERENCE OF

AMERICAN PIANO TECHNICIANS

MEETING IN CHICAGO, U. S. A.

Whose valuable and exhaustive discussions

mark an epoch in the development of

American musical technology.

This Book

is, by one who has the honor of

membership in that Conference,

RESPECTFULLY AND AFFECTIONATELY

DEDICATED

PREFATORY NOTE          (TABLE OF CONTENTS)

In writing this book, I have tried to do two things which are always thought to be admirable but seldom thought to be conjunctible. I have tried to set forth the theory of Equal Temperament in a manner at once correct and simple. Simultaneously I have tried to construct and expound a method for the practical application of that theory in practical tuning, equally correct, equally simple and yet thoroughly practical.

The construction of the piano has not in this volume been treated with minuteness of detail, for this task I have already been able to perform in a former treatise ; but in respect of the soundboard, the strings, the hammers and the action, the subject-matter has been set forth quite elaborately, and some novel hypotheses have been advanced, based on mature study, research and experience. Here also, however, the theoretical has been justified by the practical, and in no sense have I yielded to the temptation to square facts to theories. In the practical matters of piano and player repairing, I have presented in these pages the results of nineteen years' practical and theoretical work, undertaken under a variety of conditions and circumstances. In writing this part of the volume I have had the inestimable advantage of the suggestions and experiences of many of the best American tuners, as these have been gathered from past numbers of the Music Trade Review, the Technical Department of which paper I have had the honor to edit and conduct, without intermission, for fourteen years.

The preliminary treatment of the Acoustical basis of piano tuning may seem elaborate; but I have tried to handle the subject-matter not only accurately but also simply; and as briefly as its nature permits. The need for really accurate information here justifies whatever elaboration of treatment has been given.

I desire here to express my thanks to Mr. J. C. Miller for permission to utilize some of his valuable calculations, to Mr. Arthur Lund, E. E., for drawings of acoustical curves, and to my brother, Mr. H. Sidney White, M. E., for diagrams of mechanical details.

Most books intended for the instruction and guidance of piano tuners have been either so theoretical that their interest is academic purely; or so superficial that accuracy in them is through out sacrificed. I have tried to avoid both errors, and to provide both a scientifically correct text-book for teaching and a pocket guide for the daily study and use of the working tuner. The program has been ambitious ; and I am conscious, now that the task is finished, how far short of perfection it falls. But I think it fills a want; and I ask of all practitioners and students of the noble art of tuning their indulgence towards its faults and their approval of any virtues it may appear to them to possess.

The writing of the volume began in the winter of 1914 and was completed during the spring of 1915. Various causes have operated, however, to retard its publication; notably the sudden passing of the honored man whose encouragement and kindness made possible the publication of the other books which have appeared over my name. It is however fortunate that the successor of Colonel Bill, the corporation which now bears his name and is carrying on so successfully his fine work, has been equally desirous with me, of pushing the book to publication. A thorough rereading of the manuscript, however, during the interim, has suggested many slight changes and a number of explanatory notes, which have been incorporated with, or appended to, the text.

A new, and I hope valuable, feature is the Index, which I have tried to make copious and useful, to the student and to the tuner alike.

William Braid White.

Chicago, 1917.

Chapter I.               MECHANICS OF THE MUSICAL SCALE            (TABLE OF CONTENTS)

He who undertakes to master the art of piano tuning must have some acquaintance, exact rather than comprehensive, with that general body of knowledge known as Acoustics. This term is used to designate the Science of the phenomena known as Sound. In other words, by the term Acoustics we mean the body of facts, laws and rules which has been brought together by those who have systematically observed Sound and have collected their observations in some intelligible form. Piano Tuning itself, as an Art, is merely one of the branches of Practical Acoustics ; and in order that the Branch should be understood it is necessary to understand also the Trunk, and even the Root.

But I might as well begin by saying that nobody need be frightened by the above paragraph. I am not proposing to make any excursions into realms of thought too rarefied for the capacity of the man who is likely to read this book. I simply ask that man to take my word for it that I am going to be perfectly practical and intelligible, and in fact shall probably make him conclude that he has all along been a theorist without knowing it; just as Moliere's M. Jourdain discovered that he had been speaking prose all his life without knowing it. The only difference has been that my reader has not called it “theory.” He has called it “knowing the business.”

Anyhow, we are going to begin by discovering something about Sound. We are in fact to make a little excursion into the delectable kingdom of Acoustics. What is Sound? When a streetcar runs over a crossing where another line intersects, we are conscious of a series of grinding crashes exceedingly unpleasant to hear, which we attribute perhaps to flat tires on the wheels or to uneven laying of the intersecting trackage. The most prominent feature of such a series of noises is their peculiarly grating and peculiarly spasmodic character. They are on the one hand discontinuous, choppy and fragmentary, and on the other hand, grating, unpleasant to the hearing, and totally lacking in any but an irritant effect. These are the sort of sounds we speak of as “noise.”

In fact, lack of continuity, grating effect and general fragmentariness are the distinguishing features of noises, as distinguished from other sounds. If now we listen to a orchestra tuning up roughly off-stage, the extraordinary medley of sounds which results, may — and frequently does — have the effect of one great noise; although we know that each of the single sounds in the uproar is, by itself, musical. So it appears that noises may be the result of the chance mixture of many sounds not in themselves noises, but which may happen to be thrown together without system or order. Lack of order, in fact, marks the first great distinction between noises and other sounds.

If now we listen to the deep tone of a steamer's siren, or of a locomotive whistle, we are conscious of a different kind of sound. Here is the immediate impression of something definite and continuous, something that has a form and shape of its own, as it were, and that holds the same form so long as its manifestation persists. If, in fact, we continue to seek such sounds, we shall find that what are called Musical Sounds are simply more perfect examples of the continuity, the order and the definite character which we noticed in the locomotive whistle's sound. The more highly perfected the musical instrument, the more perfectly will the sounds evoked by it possess the qualities of continuity, order and definite form.

Continuity, persistence and definiteness, then, are the features which distinguish Musical Sounds from Noises. And there are therefore only two kinds of sounds: musical sounds and noises. Now, what is Sound? The one way in which we can know it, plainly, is by becoming conscious of what we call the Sensation of Sound; that is, by hearing it. If one considers the matter it becomes plain that without the ability to hear there would be no Sound in the world. Sound cannot exist except in so far as there previously exist capacities for hearing it. The conditions that produce Sound are obviously possible, as we shall soon see, to an interminable extent in all directions ; yet what we may call the range of audible Sound is very small indeed. We can hear so very little of the conceivably bearable material; if I may use so rough an expression.

So it becomes quite plain that Sound cannot be considered as something in itself, existing in the sounding body apart from us, but must rather be thought of as the form in which we perceive something ; the form, in fact, in which we perceive the behavior of certain bodies, which behavior could not be perceived in any other way. Sound then can be considered only from the view-point of the physical laws which govern the behavior of the bodies in question. The laws which govern that sort of behavior which we perceive as Sound, alone form the subject of Acoustics. Why we should experience these perception as Sound rather than as Light or Heat is not a question to be decided by Acoustics; it is not a problem of the natural sciences, but of Metaphysics.

Limited therefore to a strictly mechanical investigation, let us consider the production of Sound from this viewpoint. Suppose that I strike a tuning-fork against the knee and hold it to the ear. I am conscious of a sound only moderate in intensity but of persistent and quite definite character, agreeable, and what we call “ musical”. No one has any hesitation in calling this a “musical sound”. But what produces it, physically speaking? We can discover this for ourselves by making a simple experiment.

By lightly touching the prongs of the fork while it is sounding I discover them to be in a state of vibration. If I examine them under a microscope I shall perhaps be able to detect an exceedingly rapid vibratory motion. In order however to make sure of the existence of these unseen vibrations, it is only necessary to obtain a sheet of glass and smoke one surface of it by passing it over the flame of a candle. Then let a tuning fork be fitted with a very light needle point stuck on the end of one prong with a bit of wax, in such a position that if the sheet of glass be placed parallel with the length of the fork, the needle point will be at right angles to both.

Now set the fork to sounding, and hold it so that the needle point lightly touches the smoked surface. Have a second person then move the sheet of glass lengthwise while the fork is held still. At once the needle-point will trace out a continuous wavy line, each wave being of that peculiar symmetrical form known technically as a curve of sines or sinusoidal curve. By adjusting the experimental apparatus with sufficient exactness it would be possible to find out how many of these little waves are being traced out in any given time. Each of these waves corresponds to one vibration or pendulum-like back and forth motion of the fork. By examining the wavy line with close attention, we shall see that if the motion of the glass sheet has been uniform, each sinusoid is identical in size with all the others, which indicates that the vibrations are periodic, that is to say, recur at regular intervals and are of similar width or amplitude. We may therefore conclude from this one experiment that the physical producer of musical sound is the excitation of the sounding body into periodic vibrations.

Listen to the noise of the machinery in a saw mill. When the circular saw starts to bite at a piece of wood you hear a series of grating cracks, which almost instantly assume the character of a complete definite musical sound, though rough in character. As the saw bites deeper into the wood the sound becomes first lower, then higher, until it mounts into a regular song. As the saw comes out through the wood the sounds mount quite high and then instantly die away. What is the cause of this phenomenon?

The circular saw is a steel wheel with a large number of teeth cut in its circumference. Suppose there are fifty such teeth. At each revolution of the wheel, then, each tooth will bite the wood once. If the wheel revolves at the rate of say four revolutions per second, it follows that there will be four times fifty or two hundred bites at the wood in this time. That means that the wood will receive two hundred separate scrapes per second. Hence, the rotation of the wheel will be slightly interrupted that number of times in one second. Hence, again, the surface of the air around the wheel will be vibrated back and forth just as many times, because the entry and emergence of each tooth will cause an alternate compression and suction on the air around it. Try another experiment. Stand five boys up in a row one behind the other, so that each boy has his outstretched hands upon the shoulders of the boy in front of him. Push the last boy. He falls forward, pushes the next and regains his position. Next falls forward, pushes Third and regains his position. Third falls forward, pushes Fourth and regains his position. Fourth falls forward pushes Fifth and regains his position. Fifth has no one in front of him and so falls forward without being able to regain his position. In this way we illustrate the compression and rarefaction of the air by the alternate fallings forward and re-gainings of position undertaken by the boys. The air is even more elastic than the boys and so forms these waves of motion which we saw traced out by the stylus on the tuning fork.

Now, it is plain that as the rotation of the circular saw increases in speed the pulses become sufficiently rapid to fuse into one continuous musical sound. If the saw were rotated at irregular, constantly shifting speed, the separate shocks would not coalesce and we should have merely the sensation of a discontinuous, fragmentary, grating series of shocks which we should call a noise. Thus again we see that regularly recurring motions of the sounding body are requisite to produce musical sounds.

Transmission of Sound.

But the illustration of the five boys (which is due to the late Professor Tyndall, by the way) shows something further. It shows first how the excitation of a body into vibration at regular intervals produces an effect upon the immediately surrounding air, causing it in turn to oscillate back and forth in pulses of alternate compression and rarefaction. But it shows more. It shows that the sound-motion, as we may call it, is transmitted any distance through the air just as the shock started at one end of the row of boys is felt at the other end, although each boy moves only a little and at once recovers his position.So also each particle of air merely receives its little push or compression from the one motion of the tuning-fork or string, and transmits this to the next one. At the backward swing of the tuning-fork or string the air particle drops back to fill up the partial vacuum it left in its forward motion, whilst the motion transmitted to the second particle goes on to the third and to the fourth and so on to the ear of the hearer. Yet each particle has merely oscillated slightly back and forth. Now, this mode of transmission evidently depends upon the existence of an atmosphere. In fact, we can soon show that, apart from all question of ears, Sound could not exist for us, as we are in this state of existence, without an atmosphere. Let an alarm-clock be set to ringing and then placed under the glass bell of an air-pump.

We now begin to displace the air therefrom by working the handle of the pump. As the quantity of air inside the bell thus becomes smaller and smaller, the sound of the alarm clock's ringing becomes fainter and fainter, until, where the air is at a certain point of rarefaction, it entirely disappears; although the clapper of the alarm will still be seen working. In other words, there must be an atmosphere or other similar medium, like water, for transmission of the sound-motion from the excited body to the ear.

Properties of Musical Sounds.

Having arrived at this point, we are now in a position to discuss musical sounds in general and to discover the laws that govern their behavior. The first principle we shall lay down is that musical Sounds are distinguished from noises by the continuity of their sensation; or in other words, musical sounds are evoked by periodic vibrations. It is thus possible to measure the frequency of vibration that evokes a sound of some given height; in other words to determine its pitch. It is also possible, as we shall see, to determine a second quality of musical sounds; namely, their relative loudness or softness, or, as we shall call it, their intensity. Lastly, we can discover differences in character or quality between musical sounds, and we shall see also that it is possible to measure these differences accurately.

Loudness.

Let us begin with the second quality mentioned; that of loudness or intensity. If a tuning-fork be excited by means of a violin bow and then examined through a microscope while its motion persists, it will be observed that as the sound dies away, the amplitude or width of swing of the prongs is becoming less and less, until the cessation of motion and of the sound occur together. If, whilst the sound is thus dying away the fork is again bowed, the amplitude of the prong's motion again is seen to increase just as the sound increases. In fact, it has been found by authoritative experiments that not only does the loudness of a sound vary with the amplitude of the vibrations of the sounding body; but exactly as the square of the amplitude. For instance, if a piano string can be made to vibrate so that the width of swing in its motion is one-fiftieth of an inch, and if another piano string giving the same pitch can be made to vibrate with an amplitude of one twenty-fifth of an inch, then the second will have an amplitude twice that of the first and its sound will be four times as loud.

However, let it be remarked that the mechanical operations thus described do not necessarily correspond with what we actually seem to hear. In other words, the sensation of loudness and the mechanical cause thereof do not always agree, for the reason that we do not hear some musical sounds as well as others. For instance, it is well known that low sounds never seem as loud as high sounds, even though the amplitude of vibration in each case be the same. A low sound always sounds softer than it really should be, to use a rough expression, and a high sound louder than it really should be.

There is only one more important point about sound-intensity, namely, that the loudness of a sound varies inversely as the square of the distance of the sounding body from the hearer. Thus, other things being equal, a sound heard at a distance of fifty feet should be four times as loud as one heard at a distance of twice fifty, or one hundred feet. However, it must also be remembered that the situation of the sounding body and of the hearer in proximity to other objects, has a modifying effect upon the loudness of sound as perceived. In fact, we shall see that this is only part of the truth expressed in the term “resonance,” about which we shall have something to say later on.

Pitch.

Without making any special attempt at producing an ideal definition of “pitch,” it will be enough to call it the relative acuteness or gravity of a musical sound. Everybody knows what is meant by saying that a musical sound is high or low. The province of Acoustics lies in finding some measuring-rule, some standard, whereby we can measure this lowness or highness of a sound and place it accurately in relation to all others. The whole system of music is built upon simply a measure of pitch, as we shall see.

Now, first of all, let us find out what it is that makes a sound high or low. In other words, what is the mechanical reason for a sound producing a sensation of highness or lowness?

Musical sounds are produced through the periodic continuous vibration of some body. In the experiment of the circular saw, to which I directed attention some pages back, it was pointed out that as the speed of the saw increases, so the musical sound produced through its contact with the wood rises in height. This may be verified by any number of experiments that one chooses to make, and the net result is the fact that the pitch of musical sounds depends upon the number of vibrations in a given unit of time performed by the sounding body. Let us put it in a formula, thus:

The pitch of a musical sound varies directly as the number of vibrations per unit of time performed by the sounding body: the greater the number of vibrations, the higher the pitch.

Unit of Time.

It is customary to assign the second as the unit of time in measuring frequency of vibrations, and in future we shall use this always. If, therefore, we speak of a certain pitch as, say, 500, we shall mean 500 vibrations per second.

Double Vibrations.

In counting vibrations, we understand that a motion to and fro constitutes one complete vibration. A motion to or fro would be merely a semi-vibration or oscillation. In the United States and England it is customary to imply a double vibration (to and fro) when speaking of a “vibration.”In France the single or semi-vibration is the unit of measurement, so that the figures of pitch are always just double what they are as reckoned in the English or American style.

Range of Audibility.

It is found as the result of experiment that the human sense of hearing is distinctly limited. The lowest tone that can be distinctly heard as a musical sound is probably the lowest A (A-₁) of the piano which, at the standard international pitch, has a frequency of 27.1875 vibrations per second. Sounds of still lower frequency may perhaps be audible, but this is doubtful, except in the cases of persons specially trained and with special facilities. In fact, any specific musical sounds lower than this probably do not exist for human beings, and when supposed to be heard, are in reality not such sounds at all, but upper partials thereof.¹ The 64-foot organ pipe, which has occasionally been used, nominally realizes tones lower than 27 vibrations per second, but these are certainly not audible as specific separate sounds. They can and do serve perhaps as a bass to reinforce the upper partials of the pipe or the upper tones of a chord; but they do not appear as separate sounds, simply because the ear does not realize their pulses as a continuous sensation, but separates them. In fact, we may feel safe in concluding that the lowest A of the piano is the lowest of musical sounds generally audible. This statement is made in face of the fact that the sound evoked by the piano string of this note is usually powerful and full. This only means, however, that the sound we hear on the piano is not the pure fundamental vibration of 27.1875 vibrations per second, but a mixture of upper partials reinforced by the fundamental. Of these partials we shall have to speak later, for they are of vital importance to the due consideration of our subject-matter.²

A similar limitation confronts us when we come to the highest tones audible by the human ear. Here again there is considerable diversity of opinion as well as of experience. The highest note of the piano, C7, has a frequency of 4,138.44 vibrations per second at the international pitch. -.J^ However, there is no special difficulty in hearing sounds as much as two octaves higher, or up to 16,554 vibrations per second. Above this limit, comparatively few people can hear anything, although musicians and acousticians have been able to go much higher.³

The Musical Range.

The limits of audibility therefore embrace eleven octaves of sounds, but the musical range is considerably smaller. The modern piano embraces virtually the complete compass of sounds used in music, and, as we all know, that range is seven octaves and a minor third, from A-₁ to C₇. Let it be noted that if the range of bearable sounds lies between, say 27 and 32,000 vibrations per second, the number of possible distinct musical sounds is enormous. We know that it is quite possible for the trained ear to discriminate between sounds which, at the lower end of the gamut anyhow, are no more than 4 vibrations per second apart. For many years the late Dr. Rudolph Koenig of Paris, one of the most gifted acousticians the world has ever known, was engaged in the construction of a so-called Universal Tonometer, consisting of a superb set of one hundred and fifty tuning forks, ranging in frequency from 16 to 21,845.3 vibrations per second. In this remarkable instrument of precision, the lowest sounds differ from each other by one-half a vibration per second, while within the musical compass the difference never exceeds four vibrations. It can readily be seen therefore that the number of possible musical sounds is very much greater than the eighty-eight which comprise the musical gamut of the piano.

Just how the musical scale, as we know it, came to be what it is, I cannot discuss here; for the simple reason that the whole question is really to one side of our purpose.¹ Whatever may be the origin of musical scales, however, we know that the diatonic scale has existed since the twelfth century, although the foundation of what we call modern music, employing the chromatic tempered scale, was rightly laid only by Sebastian Bach, who died 1750. Music is a young, an infantile, art, as time goes.

The Diatonic Scale.

We have already seen that the musical tone is a fixed quantity, as it were, being the sensation that is produced or evoked by a definite number of vibrations in a given time. This being the case, it becomes evident that all possible tones must bear mathematical relations to each other. As long ago as the sixth century B.C. the Greek philosopher and scientific investigator Pythagoras propounded the notion that the agreeableness of tones when used with each other is in proportion to the simplicity of their mathematical relations. Now, if we look at the scale we use to-day we find that although the relations of the successive members of it to each other appear to be complex, yet in fact these are really most simple. Let us see how this is :

Unison.

We all know that we can recognize one single tone and remember it when we hear it a second time. If now we draw the same tone from two sources and sound the two tones together, we find that they blend perfectly and that we have what we call a Unison. If we were to designate the first tone by themathematical symbol 1, we should say that the Unison is equivalent to the proportion 1 : 1. This is the simplest of relations ; but it is so because it is a relation between two of the same tone, not between two different tones.

Octave.

We all recognize also the interval which we call the octave and we know that in reality two sounds an octave apart are identical, except that they exist on different planes or levels. So, if we play the sound C and then evoke the C which lies an octave above, we find that we have two sounds that actually blend into one and are virtually one. When we come to discover the relations between two sounds at the octave interval, we find that the higher sound is produced by just twice as many vibrations in a given time as suffice to produce the lower sound, and so we can express this octave relation mathematically by the symbol 1 : 2. This is a relation really as simple as that of the Unison, for in reality the Octave to a given tone is simply a Unison with one member thereof on a higher or lower plane.

Perfect Fifth.

The relation next in simplicity should naturally give us the next closest tone-relation. And we find that the ear at once accepts as the next closest relation what is called the Fifth. If one strikes simultaneously the keys C — G upwards on the piano one observes that they blend together almost as perfectly as the tones C — C or G — G, or any other octave or unison. The Interval or relation thus sounded is called a Perfect Fifth. When we come to trace up its acoustical relations we find that a tone a Fifth above any other tone is produced by just one and a half times as many vibrations in a given time as suffice to produce the lower tone. Thus we can place the mathematical relation of the interval of a Perfect Fifth as 1 : 1 1/2, or better still, for the sake of simplicity, as 2 : 3, which is the same thing. So we now have the simplest relation that can exist between different tones; the relation of the Perfect Fifth or 2:3. This important fact will lead to essential results, as we shall see.

The Natural Scale.

This interval, the Fifth, will be found competent to furnish us with the entire scale which the musical feeling and intuition of men have caused them, throughout the entire Western World at least, to accept as the basis of music and of musical instruments ; that is to say, the diatonic scale. If we begin with the tone C at any part of the compass and take a series of Fifths upwards we shall arrive at the following scale :

C G D E B F sharp.

These tones of course are spread over a compass of five octaves, but if they are drawn together into the compass of one octave, as they may rightly be drawn (see supra ''The Octave") then we shall have a scale like this :

C D E F sharp G A B

Now the F sharp in the present case is not actually used, but instead we have F natural, which in fact is drawn from the interval of a Perfect Fifth below the key-tone C. The reason for this preference of F natural over F sharp lies in the fact that the diatonic scale is thereby given a certain symmetry of sound which otherwise it would lack and because the work of practical musical composition is advantaged by the substitution.¹

The Diatonic Scale.

We have arrived now at the Diatonic Major Scale and although we need not here be concerned with the origin thereof, we may be satisfied to know that it appears to satisfy the musical needs of civilized mankind. Let us again examine the series of tones, this time including the octave to C, whereby we in reality complete the circle of Fifths, as it may be called, and return to the key-tone, for the octave is the same for musical purposes as the Unison. We have then, counting upwards,

C D E F G A B C

which we can readily identify as the series of seven white keys on the piano; with the eighth following and beginning a new series or scale. The complete diatonic scale, when founded on the tone C, may thus be seen, merely by looking at the piano, to consist of a series of such scales, seven in all, following one another from one end of the piano to the other.¹

Relations.

Now, if we go a step further and discover the relations which these tones hold to each other mathematically, when brought together into one octave, we find them to be as follows, expressing the lower C as 1 and the upper C as 2, and counting upwards always:

Or in other words, the relation C to D is the same as the ratio 8 to 9. The relation C to E is likewise 4 to 5. The relation C to F is 3 to 4, C to G is 2 to 3, C to A is 3 to 5, C to B is 8 to 15 and C to its octave is 1 to 2.

Tones and Semitones.

Now if we glance at the C scale as shown on the white keys of the piano we shall see that it exhibits some interesting peculiarities. Between each pair of white keys, such as C — D or D — E, is a black key, which most people know is called a sharp or a flat. But between E — F and B — C is no space whatever, these pairs of white keys being immediately adjacent to each other. If we run over the keys to sound them we shall find that the sound-interval between E — F or B — C can at once be heard as being closer or narrower, as it were, than the sound-interval between A— B or C— D or D— E, or F— G or G— A or A — B. The longer intervals, between which we find the black keys, are called Diatonic Whole Tones, and the shorter intervals E — F and B — C are called Diatonic Semitones.

Diatonic Relationships.

The exact relations subsisting between the steps or degrees of the Diatonic Scale can be ascertained by dividing the ratios previously had, by each other, pair to pair. Consulting the table previously given (page 25) showing the relations of the steps to their keytone, we find that when the ratios are divided pair by pair we get the following relations between each pair of notes :

Now the first thing that will be observed is that there are three intervals here, not two. There are in fact, evidently two kinds of whole-step or whole-tone. For it is evident that the sound-distance between C and D is more than the sound distance between D and E. In actual fact, these two whole-steps must be recognized as distinct. This, however, brings about an entirely new condition and one quite unsuspected. For inasmuch as the Diatonic Scale must of course always retain the same relationships among its successive steps, it is evident that this idea of two different kinds of whole-step must land us in difficulties.

The trouble is that we cannot always play in the key of C, by which I mean that sometimes, in fact very often, we desire to build our music upon Diatonic Scales which are founded upon other tones than C. From the point of view towards which I am leading — namely, that of tuning — we see here a serious difficulty, for it is at once evident that if we undertake to tune a Diatonic Scale, as suggested some time back, by considering it as a series of Perfect Fifths, we shall find ourselves in deep water as soon as we quit the key of C. Let me make this plainer.

Understand first of all that we have as yet talked only of a scale founded on C and therefore including what are known simply as the white keys or natural notes. Suppose we begin by tuning a series of fifths quite perfect from some given C, say for the sake of convenience a C of which the pitch is 64 vibrations per second. This is a little less than the pitch of C would be at the International standard but is more convenient for purposes of calculation. Then we should get a result like this:

Now, let us reduce this down to one octave, by transferring the higher tones down, through the simple process of dividing by 2 for each octave of transference down and multiplying by 2 for each octave of transference up. This will give us the following result:

Gathering this together, we have the following scale founded on C = 128, or in acoustical notation C2 = 128¹ .

Now, suppose we want to play a tune based on another key-tone than C. Suppose, for instance, that we want to use D 144 as the basis of a scale; that is to say, we want to play in the key of D, as we say. The first thing to do is to find out whether we have notes tuned already which will give us such a scale. Going on the same plan as the pattern Diatonic Scale of C, and applying it to D, we find that we need the following notes :

D E F# G A B C# D

All of these we already have except F# and C#. We can get F# by tuning a perfect Fifth above B 243, which will give us F# 364.5. Dropping this an octave we have F# 182.25. C# is a Perfect Fifth above F# and so will be 546.75, or, dropping an octave, 273.375. Now, we can construct a scale of D as follows, beginning with the D 144 that we already have, using all the other notes already provided and the two new ones besides. That gives us:

If you will look at it closely you will see that there must be something wrong. The distance between F# and G seems small, and so does the distance between C# and D. To test the thing, let us now construct a diatonic scale on the ratios we know to be correct and see what results we get. It works out as follows :

Now, just for purposes of comparison, let us put these two scales together, one below the other. They look like this :

At once it can be seen that the F# and the C# which we manufactured by the perfectly legitimate method of tuning perfect Fifths from the nearest tone available in the scale of C, are both wrong when secured in this way. Also, it can be seen that the B which belongs to the scale of C will not do for the scale of D. Not only is this so, but if the experiment is made with other key-tones, it will be found that they all, except the scale of G, differ somewhere and to a greater or less extent from the scale of C, even with reference to the notes which they have in common with C.

True Intonation.

It is evident, therefore, that no method of building up diatonic scales by tuning pure intervals, will do for us if we are going to use the same keys and the same strings for all the scales we need. It is evident, in fact, that if we tune perfect Fifths or any other intervals from C or any other key- tone and expect thereby to gain a scale that will be suitably in tune for all keys in which we may want to play, we shall be disappointed. Not only is this so, but it must be remembered that so far we have not attempted to consider any of the so-called sharps and flats, except in the one case where we found two sharps in the scale of D, properly belonging there. It turns out, however, when we investigate the subject, that the sharp of C, when C is in the scale of C, is quite a different thing, for instance- (as to pitch), from the C# which is the leading tone of the scale of D.

Chromatic Semitone.

The chromatic semitone, which found its way into the scale during the formative period of musical art — mainly because it filled a want — is found upon investigation to bear to its natural the ratio 24/25 or 25/24, according as it is a flat or a sharp. In the case we have been considering, then, whilst C# as the leading tone in the scale of D has a pitch, in true intonation, of 270, the C# which is the chromatic of C 256 (see previous tables) would have a pitch of 256 X 25/24 or 266.66. Similar differences exist in all cases between chromatic and diatonic semitones, thus introducing another element of confusion and impossibility into any attempt to tune in true intonation.

Derivation of chromatic ratio.

Actually the chromatic semitone is the difference between a 10/9 ratio whole tone or minor tone as it is often called, and a diatonic semitone ; thus 10/9 divided by 16/15 = 25/24.

The Comma.

The difference between the 9/8 (major) and the 10/9 (minor) tones is called a comma = 81/80. This is the smallest musical interval and is used of course only in acoustics. (9/8 divided by 10/9 = 81/80).¹

Musical Instruments Imperfect.

The above discussion, then, leads us to the truth that all musical instruments which utilize fixed tones are necessarily imperfect. As we know, the piano, the organ and all keyed instruments are constructed on a basis of seven white and five black keys to each octave, or as it is generally said, on a 12-to-the-octave basis (13 including the octave note). If now we are to play, as we of course do play, in all keys on this same key-board, it is evident that we cannot tune pure diatonic scales. The imperfection here uncovered has, of course, existed ever since fixed-tone musical instruments came into being. The difficulty, which has always been recognized by instrument builders and musical theorists, can be put succinctly as follows:

The piano and all keyed instruments are imperfect, in that they must not be tuned perfectly in any one scale if they are to be used in more than that one scale. Hence a system of compromise, of some sort, must be the basis of tuning.

The violins and violin family, the slide trombone and the human voice can of course sound in pure intonation, because the performer can change the tuning from instant to instant by moving his finger on the string, modifying the length of the tube or contracting the vocal chords. When they are played, however, together with keyed instruments, the tuning of these true intonation instruments is of course modified (though unconsciously), to fit the situation.

All tuning imperfect.

All tuning, therefore, is necessarily imperfect, and is based upon a system called “Temperament.” This system is described and explained completely in the third and fourth chapters of this book.

Temperament.

I have taken the reader through a somewhat lengthy explanation of the necessity for Temperament on the notion that thereby he will be able to understand for himself, from the beginning, the necessity for doing things that otherwise would seem illogical and inconsistent. The peculiar kind of tuning that the piano tuner must do would seem in the highest degree absurd if the student did not understand the reasons for doing what he is taught to do. Seeing also that this correct knowledge is seldom given by those who teach the practical side of the art, I thought it better to go into some detail. In any case, it is well to realize that no man can possibly be a really artistic piano tuner unless he does know all that is contained in this chapter and all that is contained in the next three. It is worth while therefore to be patient and follow through to the end the course of the argument set forth here.¹

1 See Chapter II.