The Theory of Music

Preface

This work is not so much intended for self-instruction, as to provide a systematized course by winch teachers may assist their pupils toward that general knowledge which is necessary to the true musician. The keen competition in the musical branches at present, is gradually leading the musician to become merely a specialist; to counteract this, a study of those principles which underlie all music, such as the laws of Acoustics, musical form, etc., are most necessary, and this little volume is intended to become a text-book of such studies. It naturally gives but an outline, the details of which are to be filled in by the teacher. In the matter of Acoustics, only those facts which are of the utmost importance to the musician are stated, and no attempt is made to touch upon all the ramifications of the interesting subject. It is as a means therefore, not as an end, that I hope this work may be accepted by musician and public.

Louis C. El scot.

THE

Chapter I

ACOUSTICS . STRING VIBRATIONS .

Sound is a sensation made on the organs of hearing by peculiar vibrations of the air. Acoustics treats of the laws governing these vibrations and their production, the name coming from the Greek verb "to hear." Without the presence of air there could be no sound, and the air is generally set in motion by the collision of two bodies, and the subsequent vibration of one or both of them. Such vibration can be most readily understood in the case of stretched strings, and as some of the most important musical instruments are founded on this principle we will examine these first.

Let the following figure represent a string fastened at both ends : —

If this string be plucked in the centre it will begin to

THEORY OF MUSIC.

swing to and fro on each side of its point of rest. A complete swing, first to one side and then to the other and back to the starting point is called a vibration,* as illustrated by the following figure :

Both strings and other substances have more complex vibrations which will be explained later. These vibrations of the string are imparted to the air, and thus earned to the ear. If they are very slow they are inaudible, or convey only single, isolated shocks, but when they reach sixteen per second, a tone is heard. The beauty or ugliness of a sound depends chiefly upon the regularity or irregularity of these vibrations. If they are regular and continuous a musical tone is heard, while their irregular succession will produce a noise.

The quicker the vibrations the higher the pitch : thus at about sixteen vibrations per second we hear the deepest pedal tone of the largest church organs, at thirty-two vibrations we have reached about the

* In France, the motion to each side is called a vibration, thus giving twice the number of vibrations stated in this work. Chladni used this system. The double vibration system used in England and America, originated with Newton.

TIIEOBY OF MUSIC. 9

lowest C of a seven-octave pianoforte, at 260 vibrations per second we attain a tone approximate to the middle C of the instrument, at about 4160 vibrations per second we hear the highest C of the piano, and at about 38,000 vibrations per second sound vanishes altogether, although the vanishing point is different in different ears. The speed of sound depends in some degree upon the state of the atmosphere, being quicker in warm, damp weather than in cold, clear air, but it is about 1100 feet per second in a moderate temperature.

All kinds of sound have the same velocity.

The vibrations of strings are subject to four laws which are called the Canons of the stretched string. All of these, except the one relating to tension, apply in some degree to other vibrating substances, therefore they are most important points to memorize, if the student would comprehend the construction of different musical instruments. These laws relate to the length, thickness, tension, and density of the string.

First Canon. The longer a string the slower its vibrations, in inverse proportion to the length. That is, if a string of twenty feet should have fifty vibrations per second, the same string, under the same conditions, but shortened to ten feet, would have one hundred vibrations in the same time, sounding an octave nigher than the note first produced, since twice the number of vibrations will produce the octave above a given note, and half the number the octave below. The above rule applies also to the length of organ pipes.

Second Canon. The thicker a string the slower its vibrations in inverse proportion to the thickness. That is, a string of a certain diameter will vibrate twice as slowly as one of half its diameter, all other conditions being equal.

Third Canon. The tighter a string is drawn, the more rapid will be its vibrations, in a proportion of four to one. Thus, if a string with a twenty-five pound weight at the end of it had fifty vibrations per second, by changing the weight to one hundred pounds we would attain one hundred vibrations per second, twice the original number.

Fourth Canon. The greater the specific gravity of the substance of which the string is composed, the slower its vibrations. Thus two strings of wire, the one made of platinum, the other of iron, but in all other conditions alike, will not vibrate at the same rate since platinum is nearly three times as heavy as iron. One quarter the weight of a given string will double the number of its vibrations, if other things (tension, length, etc.) remain equal.

All of the above rules are applied in the stringing of a piano, for while the upper wires are short, thin, tightly drawn, and of steel only, the lower ones are long, thick, rather loosely drawn, and coiled around with copper or wrought iron, both of which substances have a greater specific gravity than steel. In stringing the violin, the last three canons are regarded, and thickness, tension, and density (the last produced by coiling the G string) give the requisite proportion of vibrations.

Chapter II

PERCEPTION OF SOUND. OVERTONES.

We have already stated that the perception of sound by the human ear begins at about sixteen vibrations per second and ceases at about 38,000 vibrations per second ; this interval comprises about eleven octaves and a minor third, beginning an octave below the deepest C of the piano, and ending nearly three octaves and a minor third above its highest note — the five-lined C. Women are said to be able to perceive sounds which are too high in pitch to be observed by the male ear. In practical music only about seven of these octaves are employed, and although the organ has tones both higher and lower than these seven octaves, they are not used melodically, but only in combination with other notes, to give fulness, or to add the harmonics (of which hereafter) to a chord. Stated in vibrations, the lowest tone ordinarily employed in music would have about thirty-two per second, and the highest a trifle over 4000. The deepest tones attainable by an orchestra would be the following : —

THEOBT OF MUSIC.

Contrabass, four-stringed, . . EH

Contrabass, five-stringed, .

Harp,

Contrabassoon, . . p&E

&va Jos:

i

[ 'w I ova while the highest tones of Violin (harmonics) or

Piccolo are a little below

I

The vibrations of the strings move so slight a volume of air that it is always necessary to re-enforce them, just as the light of a locomotive or of a lighthouse is strengthened by reflectors. The reflector in the case of musical instruments is the sounding-board, and its function is quite as important as that of a Fresnel lens in a great lighthouse, for the sound of a vibrating string, unaided, would be inaudible at the distance of even a few feet ; it is the amplified and reflected vibrations of the sounding-board which cause the full tone which we hear when a note is sounded on piano, violin, and in a less degree, harp. The piano sounding-board is generally made of Spruce wood, and the violin makers use Pine for this important part of the instrument.

Vibrations which cause sound can differ in rapidity, extent, and shape. Difference in rapidity would affect the pitch only : difference in extent would affect the fullness and power of the tone.

E

C

The string A B, for example, will produce a louder tone when it swings as far as E, than when it only reaches D, since it gives a more decided shock to the air, producing a greater compression and expansion, and, therefore, sound-waves of a more definite and clearer character.

Differences in shape of vibrations cause differences in the quality of tones, and in these we must seek the cause of much that makes a musical tone agreeable. These differences can be best illustrated by studying the shape of string vibrations, although they exist in all vibrating surfaces.

Just as a large wave on the ocean bears smaller waves upon its breast, and these again carry yet smaller ones upon their surfaces, so a vibrating surface carries smaller vibrations along with its chief or fundamental one ; but, different from the undulations of water, these smaller vibrations subdivide in a regular order, and, as each set produces a fainter, higher tone which mingles with the fundamental tone produced by the chief vibration, it follows that a regular set of tones, each one fainter and higher than the other, is sounded simultaneously with the fundamental tone. The laws which govern the production and order of these overtones, or harmonics, as they are frequently called, were first thoroughly formulated and explained by Helmholtz, in 1863, although the fact of the existence of such tones was stated by Mersenne as early as 1636. The overtones are present in every musical tone, and it is their proportion which causes tones of the same pitch and power to be distinguishable from each other because of their quality, as for example, the tone of a violin from a flute, or a clarinet from an oboe. The character of every musical sound depends on the number and proportionate strength of the overtones, as all fundamental tones have the same quality. Nature does not present simple, or merely fundamental , tones to our ears. The tones in music which are most devoid of overtones are those produced by the larger stopped pipes in a church-organ, the lower tones of a simple keyed wooden flute, and the sound of a tuning-fork as it is dying away. When on the other hand the overtones are present in very large proportion, the result is a thin penetrating tone, such as is produced by the oboe or violin.

The effect of the overtones may therefore be summarized as follows, — a tone in which few harmonics are present is dull, hollow, and monotonous, while one with many harmonics is bright, penetrating, and sharp. A thin metallic tone can be made more mellow by causing some of the upper harmonics to disappear, and a dull and hollow tone can be made brighter by adding harmonics to it. The principle has been directly applied in the manufacture of musical instruments, and the Boehm flute was made mel> lower in tone-color, by changing its bore from conical to cylindrical, thus eliminating half of the overtones, while the organ, which was too dull toned in its diapasons, was enriched and made brighter by the addition of certain stops as the "Mixture," the " Quint," the " Fifteenth," and others, the function of which was to furnish the absent overtones.

We have said that the subvibrations which cause the overtones can be most readily observed in vibrating strings : the chief vibration would be of the entire length of the string from end to end, but smaller vibrations will instantly arise and these will sound the overtones on which the beauty of the tone will depend. The following diagrams will illustrate the manner in which these arise.

•/theory of music.

W s

Third overtone.

Fig. 3.

Second Overtone.

First overtone,

Fig. 1.

Fundamental.

Combinaton of fundamental and first overtone.

The explanation of the diagram is an explanation of the overtone system. Supposing the string to be the wire which sounds great C on a pianoforte, Fig. 1 would represent the whole vibration which sounds the fundamental note; but the wire would also vibrate in two sets of subvibrations (called ventral segments, and marked " V. S." in the diagram) and, as these would be half the length of the entire string, they would vibrate twice as fast as the full vibrations, thus sounding the note an octave above the fundamental, that is small C, as in Fig. 2. Meanwhile another set of vibrations has started, dividing the string into three ventral segments, and sounding small G, — Fig. 3; another set, begun simultaneously, divides the strings into fourths — Fio-. 4 — and sounds one-lined C ; and thus the string which is intended to represent but a single tone in the musical scale, is subdividing itself into smaller and smaller segments, and sounding an entire assembly of notes, which have been called " the chord of Nature." The following would be the harmonic series of great C, and it must be borne in mind that every tone that is sounded has its own series formed of the same intervals as those represented here, u e., the first, an octave above the fundamental, the next, a fifth higher, the next a fourth, the next a major third, the next a minor third, etc.

THE OB Y OF MUSIC.

birs flea. -&•

The notes marked thus x are not exactly of the pitch of these notes as used in our scale, but nearer to them than to any other notes in our musical system. The points marked "N" in the diagram of vibrations indicate the nodes, or points of rest (partial) between the ventral segments. If a strino- be struck near, or at, the node of any particular overtone, that overtone disappears from the series, its particular subvibration having no chance to form. This fact is of importance in the excitation of strings of instruments. The piano wires are struck by the hammers at from one-seventh to one-ninth their length, thus causing two poor overtones to disappear, the sixth and the eighth of the series. The seventh, which is good but unimportant, because faint and a mere duplication of lower tones in the series, also incidentally disappears. The violin strings are excited to vibration at about one-tenth of their length, and in harp, guitar, and all instruments of plucking or percussion, even down to the kettle drums, due regard must be had to this principle of the formation of overtones, and the point of contact thereby deter-

20 THE OUT OF MUSIC.

mined. In the principle of the overtones and theii combination is found the scientific explanation of St. Paul's sentence written nearly 2000 years ago: " And even things without life giving sound, whether pipe or harp, except they give a distinction in the sounds, how shall it be known what is piped or harped?" 1 Corinthians xiv. 7. And the question put by Paul in the first century, was completely answered only in the nineteenth by Hermann Helmholtz.

CHAPTER m.

THE TEMPERED SCALE. PITCH.

In the harmonic series given previously it was mentioned that certain of the tones did not quite correspond to any of the notes of the scale used by us. As it is pre-supposed that the reader of this volume is already familiar with the elements of practical music, we will not speak of the succession of the intervals in our diatonic scale, but we may state en passant, that, while all nations of the civilized world have chosen a fixed succession of tones of one kind or another to represent their musical systems, no one has yet been able to explain why such a succession is used, and even Helmholtz has here given a rather metaphysical, and unsatisfactory attempt at explanation. The proportion of vibrations in the different intervals of our scale, and their relation to the key-note, can however be accurately determined, and it is quite possible that Pythagoras as long ago as the sixth century, B.C., explained these. When we attempt to form a scale upon the laws of natural proportions, we obtain a result of especial sweetness, but a succession rather unsuited to modulation, if the instrument is one possessing a keyboard. As each note is calculated with reference to its relation to the keynote or tonic, it may be supposed that when the keynote changes, the entire scale of true intonation is thrown somewhat out of tune, and the farther we depart from the key first used, the more out of tune the scale will be unless a re-arrangement of the intervals is made. This is in fact the case, and it will therefore readily be understood that while certain progressions and chords can be founded on natural laws, the art of modulation is a product of the human mind, and does not derive itself from any physical law.

The scale which is at present employed by all civilized nations is a compromise. The octave which must always be taken as a true interval, its upper note vibrating twice as fast as its lower, is divided into twelve equal semitones, all a trifle out of tune with the scale of nature, but none distressingly so, and this " tempered scale" as it is called, admits of the use of all the twenty-four major and minor keys with equal facility. This system is also sometimes called i < Equal temperament " and was advocated as early as the sixteenth century by Willaert, Zarlino, and others ; but it was not thoroughly adopted until J. S. Bach in his noble collection of preludes and

THEOBY OF MUSIC. 2b

fugues entitled < < The Well-tempered Clavichord " proved the practicability of the system, by writing compositions in all of the different keys, where before only a very limited number had been employed.

The deviations of our intervals, as founded on the tempered scale system, from the true intervals as given by natural laws of proportion, are chiefly as follows, reckoning upward from the key-note :

The perfect fifth is only one-fiftieth of a semitone flat.

The perfect fourth is one-fiftieth of a semitone sharp.

These are so near to the true pitch that it would require the keenest musical ear to perceive the aberration at all. Those most perceptibly out of tune are :

The major third, one-seventh of a semitone sharp.

The minor third, one-sixth of a semitone flat.

The major sixth, one-sixth of a semitone sharp.

The minor sixth, one-seventh of a semitone flat.

The minor seventh, one-sixth of a semitone flat.

The major seventh, one-eighth of a semitone sharp.

In the scale made up of true intervals there would be a distinct difference between such notes as A- sharp and B-flat, or C-flat and B-natural, but in the tempered scale these distinctions disappear. The tempered scale came into being chiefly because of the keyed instruments, for violin and voice can give the changeable intervals of the true scale with facility, and at times attain beautiful effects thereby.

In preceding sentences we have spoken of deviations of pitch of one-fiftieth of a semitone ; these are the smallest deviations which even a trained musical ear can hear. We can scientifically prove two tones to differ by one one-hundredth of a semitone, but it would not be perceived by any human ear. Ordinary ears can distinguish from fifty to 100 different intervals to the octave, and highly trained ears about 600.

It seems strange that one cannot represent a fixed tone, as for example middle C, to the mind by a set number of vibrations, but the standard of pitch has always been a variable one and the note in question might consist of more or less vibrations according as it belonged to a higher or lower standard of pitch. Thus the note A which in Paris at present has 435 vibrations, in 1858 was given 448, and in 1699 had only 404, while Handel's tuning-fork dated 1740, gives the same note 416 vibrations. The standard of pitch has been gradually rising since the days of Bach and Handel, in whose time it was about two-thirds of a tone deeper than at present. The cause of this rise was the greater brilliancy sought for by manufacturers of pianos and wind instruments ; but of course the vocalist was the sufferer by it, and in our own time it is gratifying to find a brave army battling with good success for a lower standard of mtch. The high pitch of the present is called " concert pitch," the reformed or lowered pitch is generally called the " normal," and is practically the same as the French pitch of the present time, which has received the endorsement of almost all the leading acousticians, the French government, and many orchestral directors. The differences in the standard of pitch used at present in various countries may be understood from the following list : — | (L r

English (average), 540 vibrations.

German (Scheibler), 528 vibrations.

French normal diapason, 517 £ vibration. *

New England Conservatory of Music, 517 ^vibrations.

Classical (the average pitch at the beginning of the century), 512 vibrations.

Philosophical, or physical pitch, 512 vibrations.

American concert pitch is a very doubtful quantity, but is the highest pitch used anywhere.

The classical pitch is so called because it represents the average standard employed by such composers as Beethoven, Haydn, Mozart, Schubert, Weber, etc. The pitch used by Handel and Bach was somewhat lower.

Philosophical or physical pitch is so called because it is attained by a simple, physical process. We

* In 1892 The American Piano Manufacturers' Association adopted this, "which may now be called "The International Pitch." A, 435; C, 517-^.

have already seen that halving the length ot a string loubles the number of its vibrations. If we take a long wire, and draw it tight until a tension has been attained which will allow it to vibrate once a second (a result which can be readily attained by the eye, and a regulated pendulum) we will hear no sound ; we can however call this imaginary note C, which, at one vibration per second, will be the unit of the physical system of ascertaining vibrations. If we now measure off half of the wire, we will have a length which will give us two vibrations per second ; another division of the wire and we attain four vibrations, another and we have eight, yet another and the speed of sixteen vibrations is reached, and a deep tone is heard. If we now continue dividing the wire in the same manner, at the ninth division we attain the rate of 512 vibrations per second, a note which is nine octaves above the first, theoretical C of one vibration per second. This is but one of many modes of ascertaining the number of vibrations of a given note, for although we have only ascertained the number of vibrations in the different C's, yet if the number of vibrations in any one note be known, one can by a simple arithmetical process compute the number contained in any other note. The following table will present the proportion of the vibrations in the two notes of some of the musical intervals according to the scale of true intonation (not the tempered intervals) : —

THEOBT OF MUSIC.

Octave. 2

Major seventh. 15 8

Minor seventh. 16 9

Major sixth. 5 3

Minor sixth. 8 5

Perfect fifth. 3 2

Diminished fifth. 36 25

Perfect fourth. 4 3

Major third. 5 4

Minor third. 6 5

Major second. 9

Minor second. 16 15

The logarithms of these ratios would be as follows :

Octave. 301

Major seventh. 273

Minor seventh. 250

Major sixth. 222

Minor sixth. 204

Perfect fifth. 176

Diminished fifth. 158

Augmented fourth, 143

, Perfect fourth. 125

Major third. 97

Minor third. 79

Minor second.

Major second. 51

The application of these tables, which represent the enharmonic or natural intervals, would be as follows : if middle, or one-lined C [ fa -| has 260 vibra-

tions per second, this note,|^ p^ being a perfect

fifth above it, would have the proportion of three to

two, i, e. , three-halves of 260 = 390 ; this note | fa

an octave above the last would have twice its vibrations,

i. e. , 780 ; this note [ (h ^ 1 a major third above middle

C, would have five-fourths of 260 = 325; and in the table of logarithms a fifth added to a fourth, a minor third to a major sixth, a minor second to a major seventh, etc., will give the logarithm of the octave.

In W. Pole's admirable " Philosophy of Music" a very full explanation of these proportions will be found.

CHAPTER IV. CLASSIFICATION OF A r IBRATIONS

Ix order that the student may fully understand the musical instruments which are to be described in the next chapters, it may be well to reiterate the fact that all musical sounds consist of regular vibrations , and to supplement it with the statement that these vibrations can be classified as follows : —

First. The vibrations of strings ; these differ in quality according to the manner in which the string is set in motion, i. e., by striking, as in the piano; by plucking, as with harp or guitar ; and by friction, as with instruments of the violin family.

Second. The vibrations of reeds. A reed, in musical instruments, is a thin tongue of wood or metal , against which a current of air is directed ; the reed, swaying rapidly to and fro, breaks this current into intennittent puffs, which form the tone. Reed vibrations generally produce many overtones, and therefore a rather thin and penetrating quality of sound. Cabinet-organs, oboes, clarinets, bassoons, etc., are examples of reed vibrations.

Third. The vibrations of elastic membranes, generally organs of the human body, set in motion by a current of air. The lips of a player of horn, trumpet, cornet, trombone, etc., and the vocal chords in the human throat, are examples of this kind of vibration, which is of the same general character as reedvibration.

Fourth. The vibrations of elastic membranes set in motion by a blow. These move the air in the same manner as a sounding-board would do. Drums give examples of such vibrations, and although most drums do not produce a strictly musical sound, the kettle-drums give as musical a tone as almost any percussive instrument of their pitch.

Fifth. The vibrations of solid substances of elastic material. These are almost always set in motion by percussion. Bells, tuning-forks, etc., are examples of this class of vibrations.

Sixth. The vibrations of the air upon itself in a confined space, — generally a tube. Although the air which gives the sound-vibrations is generally set in motion by some vibrating substance, yet the assistance of the latter is not always necessary. If we force a column of air with some degree of rapidity into or across a tube in such a manner that instead of passing directly through the tube it will cause a flutter of air at one end, we at once produce a tone directly from the air itself, without the necessity of any other vibrating body, for the tube will at once select the sound-waves which suit its length, or synchronize with it. The wind whistling in a chasm or cave is a natural example of this, while flutes, organ (diapason) pipes, etc., show in what manner man has applied the principle. In the flute, the breath of the player is blown at a certain angle and with a certain power or no tone will result, but in the organ-pipe this angle or deflection which causes the alternate expansion and contraction of the air necessary to produce sound-waves, is caused by a plug or sharp edge (similar to that which the boy places at one end of his whistle) , at the mouth. It remains only to add that the pitch of open pipes depends chiefly on their length. The width of pipes affects the quality of the tone. In wide pipes the harmonics are not prominent, and therefore these have hollow tones ; narrow pipes have prominent harmonics, and therefore bright and penetrating tones. The same pipe can produce more than one tone ; if the rapidity of the vibrations be increased, other tones ensue. In a cylindrical pipe or tube only every alternate overtone is strong, which gives a rather mellow quality to the tone. Flutes and clarinets are made upon this principle. In a conical pipe all the overtones are clearly present up to a reasonable height, which causes the penetrating quality heard in the tones of the oboe and bassoon.

If a pipe that is open be plugged at its end (it is then called a "stopped pipe," and the plug a " tompion,") it at once sounds an octave deeper, and the tone becomes more hollow and dull, because the column of air is now undivided, while in an open pipe a node is formed at the centre of the tube, which divides the column in two, giving both a brighter and higher sound.

The foregoing facts, although merely an outline of the connection between practical music and acoustics, may yet suffice to give a general comprehension to the student of the principles upon which musical instruments are made.

Chapter V.*

THE ORCHESTRA AND ITS INSTRUMENTS. THE VIOLIN.

We have already seen how the laws of acoustics are applied to the manufacture of pianos and organs ; we shall find the same laws applied in a similar manner to the making of each instrument of the modern orchestra. The word " orchestra " comes from the Greek, and signifies "a place for the chorus," for in