B 3 am 517 '«. 7. ^« ^•' ^ ■|..:-.^V^^ -LJl::5v. OF THK ^hiiucrsitg of ^alifffi|iiia, Division Phange Shelf Received A^-" C^\ --Ili:^- ^^ X---^' ^' j> _£-/ Let ABCDEF represent a section of a part of the sea-surface at any given instant, and suppose that during, say, the next ensuing second of time, the separate drops in ABCDEF move vertically, either upwards or downwards as shown by the arrows, so that, at the end of that second, they all occupy positions along the dotted curved line AEC'UFF\ The two portions, ABODE and B' C D' F F\ are exactly alike, and, therefore, the effect is just what it would have been had we pushed the curve ABODE along horizontally until it came to occupy the position B'0'D'E'F\ In order further to illustrate this point, let us suppose that a hundred men are standing in a line and that the first ten are ordered to kneel down : a spectator who is too far off to distinguish individuals will merely see a broken line like that in the figure below. MM 10 TRANSFERENCE OF RELATIVE POSITION. [I. § 5. Now, suppose tliat, after one second, tlie eleventh man is ordered to kneel and the first to stand ; after two seconds the twelfth man to kneel and the second to stand; and so on. There will then con- tinue to be a row of ten kneehng men, but, during each second, it will be shifted one place along the line. The distant observer will therefore see a depression steadily advancing along the line. The state of thmgs presented to his eye after two, six, and nine seconds, respectively, is shown in Fig. 2. Ilxg.2. iiiiiiiiiiiiiiiiiH There is here no horizontal motion on the part of the men composing the line, but their vertical motions give rise, in the way explained, to the hori- zontal transference of the depression along the line. The reader should observe that for no two con- secutive seconds does the kneeling row consist of exactly the same men, while in such positions as I. § 6.] LENGTH AND AMPLITUDE OF WAVE. 11 those shown in the figure, which are separated by more than ten seconds of time, the men who form it are totally different. 6. Let us now return to the sea-waves, and examine more closely the elements of which they consist. Fig. 3 represents a vertical section of one com- plete wave. Ti^.Z The dotted hne is that in which the horizontal plane, forming the surface of the sea when at rest, cuts the plane of the figure. The distance between the two extreme points of the wave, measured along this hne, is called the length of the ivave. C is the highest point of the crest DCB ; E the lowest point of the trough AED. CF and GE are vertical straight lines through C and E ; HCK and LEM are horizontal straight linea through the same two points. The vertical distance between the lines HK and LM is called the breadth or amplitude of the wave. Thus AB is the length of the wave, and, if we produce EG and CF to cut the lines HK and 12 FORM OF WA VE. [I. § 6. LM in iV^ and P respectively, we have, for its amplitude, either of the equal lines EN, PC, Each of these is clearly equal to FC and GE to- gether, that is to say, the amplitude of the wave is equal to the height of the crest above the level- line together with the depth of the trough below it. In addition to the length and amplitude of the wave, we have one more element, its form. The wave in the figure has its crest shorter than its trough and higher than its trough is deep. Moreover the part i)(7of the crest is steeper than the part CB, while, in the trough, the parts AE and EB are equally steep. Sea-waves have the most varied shapes according to the direction and force of the wind producing them. Hence, before we can lay down a wave in a figure, we must know the nature of the wave's curve, or, in other words, its form. Since the crests of the waves are raised above the ordinary level of the sea, the troughs must necessarily be depressed below it, just as, in a ploughed field, the earth heaped up to form the ridges must be taken out of the furrows. Each crest being thus associated with a trough, it is con- venient to regard one crest and one trough as form- ing together one complete wave. Thus each wave consists of a part raised above, and a part depressed below, the horizontal plane w^hich would be the I. § 7.] THREE ELEMENTS OF WAVE. 13 surface of the sea were it not being traversed by waves. 7. The length, amplitude, and form of a wave completely determine the wave, just as the length, breadth, and height of an oblong block of wood, i. e. its three dimensions, fix the size of the block. These three elements of a wave are mutually independent, that is to say, we may alter any one of them with- out altering the other two. This will be seen by a glance at the accompanying figures* Fi^ 4 (0 (1) shows variation in length alone ; (2) in am- plitiide alone ; (3) inform alone. 14 DROP- AND WA YE-MOTION. [I. § 7. I. § 8.] DROP- AND WAVE-MOTION. 15 8. We will next study more closely tlie motion of an individual drop of water, in the surface of the sea, wliile a wave passes across it. Fig. 5 sliows nine positions of the wave and moving drop at equal intervals of time, each one-eighth of the period during which the wave traverses a horizontal dis- tance equal to its own wave-length. In (1), the front of the wave has just reached the drop j)re- viously at rest in the level-line represented by dots in the figure. In (2), the drop is a part of the way up the front of the crest ; in (3), at the summit of the crest, and, therefore, at its greatest distance above the level-line. In (4), it is on the back of the crest, and, in (5), occupies its original position. It then crosses the level-line ; is on the front of the trough in (6), and at its lowest point ui (7), where it attains its greatest distance below the level-line. In (8), it is on the back of the trough, and, in (9), has once more returned to its starting-point in the level-line. We have here a vibratory or oscillatory move- ment, like that of the end, or 'bob,' of a clock- pendulum, but executed in a vertical straight line. We call the distance between the two extreme posi- tions of the bob, the extent of swing of the pen- dulum. The extent of the drop's oscillation will be seen, from (3) and (7), to be equal to the sum 16 EXTENT AND A2IPLITUDE. [I. § 8. of tlie height of the wave's crest above the level- line, and of the depth of the trough below it. But this, as was shown in § 6, is equal to the amplitude of the wave. Hence 'extent of drop's vibration' and 'amplitude of corresponding wave' are only different ways of expressing the same thino^. Let the line A'OA be that in which the drop A. under consideration vibrates, being in the level- line, A and A' the limits of oscillation. The w^hole movement given in Fig. 5 will then be from to ^, from A through to A\ and from A' back again to 0. This is termed one complete vibration, and since, ui the course of it, each portion of the drop's path is passed over tivice, one complete vibration is equal to an upward swing from A to A together with a downward swing from A to A\ In the clock-pendulum we have, during each second, one complete oscillation, consisting of one swing from left to rio;ht and one from rio-ht to left. I. §9.] VIBRATION-PERIOD AND WAVE-LENGTE. 17 Reference to Fig. 5 at once shows that, during the time occiij)ied. by the wave in traversing its own wave-length, the moving drop performs one complete vibration, or, to express the same fact in tlie reverse order, that ivliile the drop snakes one complete vibration, the luave advances through one wave-length. This is a most important principle, and should be thoroughly mastered and borne in mind by the student. 9. What has just been proved for a particular drop is, of course, equally true for any assigned drop in the surface passed over by a wave. All the drops, therefore, go through exactly the same vi- brations in exactly equal times, but, since each drop can only start at the moment when the front of the wave reaches it [Fig. 5, (1)], they will in general occupy different positions in their paths at the same time. We may illustrate this by supposing a number of watches, which keep good time, to be set going successively in such a way that the first shall mark xii at twelve o'clock, the second at five minutes past twelve, the third at ten mi- nutes past twelve, and so on. The hands of each watch will describe the same paths in equal times, but, at any assigned moment, will occupy different positions in those paths corresponding to the late- ness of their several starts. The drops in the sea- T. 2 18 SUCCESSIVE VIBRATIONS. [T. § 10. surface, being, in tliis manner, thrown successively into the same vibratory motion, give rise, by their consequent varieties of position at any assigned moment, to the transmission of the form which we call a wave. When a series of contmuous equal waves, such as those in Figure 7, are being transmitted, each oscillating drop, after completing one vibration, will instantly commence another precisely equal vibration, and go on doing so as long as the series of waves lasts. The kind of motion in which the same movement is continuously repeated in suc- cessive equal intervals of time, is called 'periodic,' and the time which any one of the movements occupies is called its 'period.' Thus, to continuous equal waves correspond continuous periodic drop- vibrations. 10. AYe will next compare the periods of the drop-vibrations corresponding to waves of different lengths advancing with equal velocities. In Fig. 8 waves of three different lengths are ref rssented. One wave of (l) is as long as two of (2), and as three of (3). Therefore a drop makes I. § 10.] VIBRATIOX-RATE AND WAVE-LENGTH. 19 one complete vibration in (l) while the long wave passes from A to By two in (2) while the shorter CO ^ (0 (0 waves there presented pass over the same distance, and three in the case of the shortest waves of (3). But the velocities of these waves being, by our sup- position, equal, the times of describing the distance AB will be the same in (1), (2), and (3). Hence a drop in (2) vibrates twice as rapidly, and a drop in (3) three times as rapidly, as a drop in (1); or con- versely, a drop in (1) vibrates half as rapidly as a drop in (2), and one third as rapidly as a drop in (3). The rates of vibration in (1), (2) and (3), (by which we mean the numbers of vibrations performed in any given interval of time) are, therefore, propor- 2—2 20 MODES OF OSCILLATIOX. [L § H. tional to tlie numbers 1, 2 and 3, which, are them- selves inversely proportional to the wave-lengths in the three cases, respectively. We may express our result thus ; the rate of drop-vihration is inversely jproportional to the corresponding icave-Iength. The same reasoning will apply equally well to any other case ; the proposition, therefore, though derived from particular relations of wave-lengths, is true univer- sally. 11. We have now connected the extent of the drop -vibration with the amplitude, and its rate with the length, of the corresponding wave. It re- mains to examine what feature of the oscillatory movement corresponds to the thiixl element, the form, of the wave.- Fig. 9. \ — 4 1 B O j^ Suj)pose that two boys start together to run a race from to A, from A to B, and from B back to 0, and that they reach the goal at the same moment. They may obviously do this in many different ways. For instance, they may keep abreast aU through, or one may fall behind over the first half of the course and recover the lost ground in the second. Again, one may be in front over OAO, and the other over OBO, or each boy may pass, and be passed by, his competitor, repeatedly during the race. We may I. §11.] VIBRATION-MODE AND WAVE-FORM. 21 regard the movement of each boy as constituting one complete vibration, and thus convince ourselves that an oscillatory motion of given extent and period may be performed in an indefinitely numerous variety of modes. Let us now compare the positions of a drop at successive equal intervals of time, when cooperating in the transmission of waves of different forms. O 1 3" 4: 5 6 re 9 10 In each of the three cases in Fig. 10 the front of a wave-crest is shown in the positions it respec- tively occupies at the end of ten equal intervals of time (each one tenth of that occupied by the wave 22 riBEATlON-MOBE AND WAYE-FOBM. [I. § 12. in traversing a quart er-wave-lengtli), the apex of the wave being successively at the equidistant points of the level line 1, 2, 3, 4, &c. A drop whose place of rest is 0, will then assume the corresponding positions in the vertical line OA : thus the points where this line cuts the successive wave-fronts show the positions of the vibrating drop at equal intervals of time. By comparing the three cases it will be seen that the mode of the drop's vibration is distinct in each. In (1), it moves fastest at 0, and then slackens its pace up to ^. In (2), it starts more slowly than in (1), attains its greatest speed near the middle of OA^ and again slackens on approaching A. In (3), the pace steadily increases from O to ^. The different waves in the figure have been purposely drawn of the same amphtude and length, in order that only such variations as were due to differences of form might come into consideration. The reader should construct for himself similar figures with other wave-forms, and so convince himself, more thoroughly, that every distinct form of wave has its own special mode of drop-vibration. 12. The converse of this proposition is also true, viz. that each distinct mode of drop-vibration gives rise to a special form of wave. We will show this by actually constructing the form of wave I. § 12.] CONSTRUCTION OF WAVE-FORM. 23 which answers to a given mode of drop-vibration. When a drop vibrates in a given mode, its position at any assigned moment during its vibration is of course known. If we also know the amount by which drops further on in the level-line are later in their starts [§ 9] than drops less advanced in that line, we can assign the positions of any number of given drops at any given instant of time. Suppose that each drop makes one complete vibration per second about its position of original rest in the level-line. The law of its vibration is roughly indicated in Fig. 11. 6 3 7- -i -o^ie 9- 10- ~15 11- 12— -14. -13 A AB is the path described by any drop ; its position when in the level line; 1, 2, 3, 4... 16 its positions after 16 equal intervals of time each one-sixteenth of a second : 16 coincides with 0, i.e. the drop has returned to its starting-point. 24 CONSTRUCTION' OF WAVE-FORM. [I. § 12. Next, select a series of drops origmally at rest in equidistant positions along the level-line, and so situated that each commences a vibration, identical with that above laid down in Fig. 11, one-sixteenth of a second after the drop next it has started on an equal oscillation. Fig. 12 shows the rest-positions of the series of drops (Xo, Ctj, (X2, Ci^,.*CtiQ, in the level-line, and their contemporaneous positions during a subsequent vibration. 'k k ■ np^. h f9 • iO ^ K j^T ^8 ^i, ^io «il «^ ^ S^^5 ^ i I ^; The moment selected for the figure is that in which the first of the series, cIq, is on the point of commencing its vibration in a vertical direction. Since the second drop started one-sixteenth of a second after the first, its position in the figure will be below the level-line at a/ making the line I. § 13.] COIS^STIWCTION OF WA YE-FORM. 25 a^ cti equal to the line 015 in Fig. 11. The next drop, which is two-sixteenths behind cIq in its path, will be at a.^ making a.^ a.^ equal to Ol4 in the same figure. In this way the positions of all the points ai a^a-i, Sec, in Fig. 12 are determined from Fig. 11. They give us, at once, a general idea of the form of the resulting wave. By laying down more points along the line AB in Fig. 11, we can get as many more points on the wave as we please, and should thus ultimately arrive at a continuous curved line. This is the wave-form resulting from the given vi- bration-mode with which we started, and, since only one wave-form can be obtained from it, we infer that each distinct mode of drop-vibration gives rise to a special form of wave. It has now been sufficiently shown that corre- sponding to the three elements of a wave, amjjli- tude, length, sudform, there are three elements of its proper drop-vibration, extent, rate, and mode. 13. We have seen that a sea- wave consists of a state of elevation and depression of the surface above and below the level-plane. The same thing holds of the small ripples set up by throwing a stone into a pond, and the non-progressive nature of the motion of individual drops on the surface can be as easily made out on a small, as on a large sheet of water. Moreover the characteristic phenomenon on which 26 EXTEXsiox OF TER2I 'Wave: [I. § u. we have been engaged, viz. a liniformly j^^^ogressive motion arising out of a numher of oscillatory move- ments, is by no means confined to liquid bodies. Thus, when a carpet is being shaken, bulging forms, exactly like Avater- waves, are seen running along it. A flexible string, one end of which is tied to a fixed j^oint, and the other held in the hand, ex- hibits the same phenomenon when the loose end is suddenly twitched. It has accordingly been found convenient to extend the term ' wave ' beyond its original meaning, and to designate as ' wave-motion ' any movement which comes under the definition just laid down. We proceed to an instance of such motion which is important from its similarity to that to which the transmission of Sound is due. 14. Any one who has looked down from a slight elevation on a field of standmg corn on a gusty day, must have frequently observed a kind of thrill running along its surface. As each ear of corn is capable of only a shght swaying movement, we have here necessarily an instance of ivave-motion, the ear- vibrations corresponding to the drop-vibrations in water-waves. There is, however, this important dif- ference between the cases, that the ears' movements are mainly horizontal, i. e. in the line of the waves advance, whereas the drop-vibrations are entirely perpendicular to that line. The advancing wave I. § 15.] LONGITUDINAL TIB RATIONS. 27 is therefore no longer exclusively a state of elevation or depression of surface, but of more tiglitly, or less tiglitly, packed ears. The annexed figure gives a JFig i3 rough idea how this takes place. The wind is sup- posed to be moving from left to right and to have just reached the ear A. Its neighbours to the right are still undisturbed. The stalk of C has just swung back to its erect position. The ears about B are closer to, and those about C further apart from, each other, than is the case with those on which the wind has not yet acted. After this illustration, it will be easy to con- ceive a kind of wave -motion in which there is no longer (as in the case of the ears of corn) any move- ment transverse to the direction in which the wave is advancing. 15. Let a series of points, originally at rest in equidistant positions along a straight line, as in (0), Fig. 14, be executing equal periodic vibrations in that line, in such a manner that each point is a 28 LOFGITUDIXAL VIBRATIONS. [I. § 15. certain fixed amount furtlier back in its path than is its neighbour on one side, and therefore exactly as much more forward than is its neighbour on the other side. (l) shows the condition of the row of points at the moment when the extreme point on the left is beginning its swing from left to right, which, in accordance with the direction of the arrow in the figure, we may call its forward swing. The equi- distant Vertical straight lines ^ the extent of vibration for each oscillating point. The constant amount of retardation between successive points is, in the instance here selected, one-eighth of the path traversed by each point during the period of a complete oscillation. Thus, proceeding from left to right along the line (l), we have the first point beginning a forward swing, the second, third, fourth and fifth points entering respectively on the fourth, "third, second, and first quarters of a backward swing, and the sixth, seventh, eighth and nmth points on the fourth, third, second, and first quarters of a forward swing. Since the ninth point is just beginning a forward swing, its situation is exactly the same as that of the first point. Beyond this point, therefore, we have only repetitions of the state of things between the first and ninth points. The row^ (l) is therefore I § 15.] LONGITUDINAL VIBRATIONS, Fig. ±4<. 29 A. B ^^^^~^\ ^-^^ -^5'- """^ ^"'^^"-^ W. Ff ® .© ^ > ® ^^ «r r- \ © © ® ^€ "® @ ®^ ^ CO a \ V \ ^ \ '^ V s> ® @ @ € e e ® » © ® fjj k k \' b ' © © © € ® ® « 6» e ©• ^. \ \ \ it k y 3 ® © ® € @ (0 € ® @ S &^ \ \ ^e '# ®^ b ^, © © € (0 d e 6 ® @ 9| '^ \ 'b e ^ \ \ b ^ ® ® @ CO e © € ® © @ ^ \ ^. h « b ^^ ^ ® © (0 © © © € © ® • \ k \ '^> ® (0 ® © © e ® ® 'I k ^ b \ > ^ E» made up of a series of groups, or cycles, of tlie same number of points arranged in the same manner throughout. Two such cycles, included by the large brackets A and B, are shown in (1). Each cycle is divided by the small brackets a, a and h, h' into 30 LOXGITUDIXAL YIBRATIOKS, [I. § 15. two parts. In a and h tlie distances between suc- cessive points are less than, and in a and h' greater than, the corresponding distances when the jDoints occupied their undisturbed positions, as in (0). The cycles correspond to complete waves on the surface of water, the shortened and elongated portions of each cycle answering to the crest and trough of which each water-wave consists. (2) shows the state of the row of points when an interval of time equal to one eighth of the period of a complete point-vibration has elapsed from the moment shown in (1). The wave A has here moved forward into the position indicated by the dotted hues. The following rows (3,) (4), (5), &c., show the state of things when two-eighths, three-eighths, four-eighths, &c., of a vibration-period has elapsed since (1). In each, the wave A moves forward one step further. In (9), a whole vibration- j)eriod has elapsed since (1). Accordingly every oscillating point has per- formed one complete vibration, and returned to the position it held in (1). The wave A, meanwhile, has travelled constantly forward so as to be, in (9), where B was in (1), i.e. to have advanced by one whole wave-length. The 2^^^02)03^1011 proved for loaves due to transverse vibrations in § 8 is thus I. § 16.] LONGITUDINAL VIBRATIONS. 31 shown to hold good likewise for ivaves due to longitu- dined vibrations. 16. In the waves shown in Fig. 14, the points in the bracket a are mutually equidistant, as are also those in the bracket h. This is due to the fact that, in the case there represented, the oscil- lating points move uniformly, i. e. with equal ve- locity, throughout their paths. If we take other modes of vibration, we shall find that this equi- distance no longer exists. Fig. 15 shows three distinct modes of vibration with the wave resulting from each, on the plan of (1) Fig. 14. The extent of vibration, and length of wave, are the same in the three cases. In (I) the points move quickest at the middle and slowest at the ends, of their paths ; in (II) fastest at the ends, and slowest m the middle; in (III) slowest at the left end, and fastest at the riofht. The shortest distance separating any two points contained in a is, in (I), that between 7 and 8 ; in (II), that between 8 and 9 ; in (III), that between 5 and 6. The corresponding greatest distances are, in (I), between 2 and 3 ; in (II), between 1 and 2 ; in (III), between 4 and 5. The remaining points likewise exhibit differences of relative distance in the three cases. Thus, the positions of greatest 32 LONGITUDINAL VIBRATIONS. [I. § 17. shortening, and greatest lengthening, occupy dif- ferent situations in the wave, and the interme- Flfj.13, (ur; I i \ 3 J 5 6 © 7 s ^ d 9 I " 6 5 6" a o 8 ® 3 4- 7 9 Jb a 7 1 © 2 e 4i ® 8 9 9 diate variations between them, proceed according to different laws, when the modes of point-vibra- tion are different. The more points we lay down in their proper positions in a and h, the less abrupt will be the changes of distance between successive points. By indefinitely increasing the number of vibrating points, we should ultimately arrive at a state of things in which perfectly con- tinuous changes of shortening and lengthening inter- vened between the positions of maximum shorten- ing and maximum lengthening in the same wave. 17. Let us now replace our row of indefinitely numerous points by the slenderest filament of some material whose parts (like those of an elastic string) admit of being compressed, or dilated, at pleasure. I. § 18.] COXBENSATIOX AND RAREFACTION, 33 When any portion of the filament is shortened, a larger quantity of material is forced into the space which was before occupied by a smaller quantity. The matter within this space is, therefore, more tightly packed, more dense, than it was, i. e. a process of condensat{o7i has occurred. On the other hand, when a portion of the filament is lengthened, a smaller quantity is made to occupy the space before occupied by a larger quantity. Here the matter is more loosely packed, more rare, than it was, i. e. a process of rarefaction has taken place. Let us now suppose the particles, or smallest conceivable atoms, of the filament, to be thrown into successive vibrations in the manner already so fully explained. Alternate states of condensation and rarefaction will then travel along the filament. It will be convenient to call these states 'pulses' — of condensation or rarefaction as the case may be. A pulse of condensation and a pulse of rarefaction to- gether make up a complete wave. 18. The degree of condensation, or rarefaction, existing at any given point of a wave has been shown to depend on the mode in which the particles of the filament vibrate. It is therefore desirable to have some simple method, appealing directly to the eye, of exhibiting the law of any assigned mode of vibration which takes place in a straight line. T. 3 34 ASSOCIATED WAVK [I- § 1^. "VVe may arrive at such a metliod by the following considerations. When a line of particles vibrate longitudinally, they give rise to waves of condensation and rarefac- tion ; when transversely, to waves of displacement on opposite sides of the line of particles in their positions of rest. Nevertheless, if the vibrations in the two cases are identical in all other respects save direction alone, the distance which, at any moment, separates an assigned particle from its position of rest will be the same, whether the vibrations are longitudinal or transverse. It is therefore only necessary to construct the wave corresponding to any system of transverse vibrations, in the way shown in § 12, to obtahi the means of fixing the position of an assigned particle, at any given moment, for the same system of vibrations executed longitudinally. Let AB and CD, Fig. 16, be lines of particles executing vibrations transverse to AB, and along CD, I. §19.] ASSOCIATED WAVF. 35 respectively. Let a and h be corresponding particles in their positions of rest. Draw the transverse wave for any given instant of time : the particle originally at a will now be at a, and that originally at h, at h\ making hb' equal to aa. By performing the same process for different instants, we can find as many corresponding po- sitions of the longitudinally vibrating particle as we please. It is true that we learn nothing new by this, since we cannot construct the wave-curve Avithout knowing beforehand the mode of the par- ticle's vibration [§ 12]. Still when we are dealing with longitudinal particle-vibrations, and require to know the law of the variation of condensation and rarefaction at different points of a single wave, it is convenient to have a picture of the mode of vibration by which, as we know [§ 16], that law is determined. Such a picture we have in the form of the wave produced by the same mode of vibration when executed transversely. Let us call the wave so related to a oiven wave of con- densation and rarefaction, its associated ivave. 19. Before leaving this portion of the subject, it will be advisable to draw the associated wave for that particular mode of longitudinal vibration in which each particle moves as if it v/ere the extremity of a pendulum traversing a path which is very short 3—2 36 PEXD UL UM-VIBEA TIOX. [I. § 19. compared to tlie pendukim's length. The meaning of this limitation will be easily seen from Fig. 17. Let be the fixed point of suspension ; OA the pendulum in its vertical position ] AB ^ portion of a cu-cle with centre and radius OA ; a, h, c, d, points on this circle ; AD ^ horizontal straight line JF15.17. through A ; act, W , cc', del' verticals through a, h, c, d, respectively. If the pendulum is placed in the position Oa, and left to itself, it will swing through twice the anoxic aOA before it turns back ao^ain. Similarly if started at Oh, it will swing through twice the angle hOA; if at Oc, through twice the angle cOA, and so on. Now, the extremity of the pendulum, when at a, is further from the horizontal line, AD, than when it is at h, since aa is greater than 1)1), and at h further than at c. If we make I. §19.] PENDULUM-VIBRATION, 37 the pendulum vibrate through only a small angle, by starting it, say, in the position Od, its extremity will, throughout its motion, be very near to the horizontal straight line AD. If we make the angle small enough, or, which is the same thing, take Ad sufficiently small compared with OA, we may, without any perceptible error, suppose the end of the pendulum to move in a horizontal straight line, instead of in a circular arc, i. e. along d'A instead of dA. To take an actual case, let us suppose the pendulum to be 10 ft. long, and that its extent of swing is 1 inch on either side of its vertical position. A very easy geometrical calculation will show that the end of the pendulum will never be as much as th of an inch out of the horizontal straio^ht hne 200 ° drawn through it in its lowest position. This is a vanishing quantity; we may, therefore, safely regard the vibration as performed along d'A instead of dA. Such a vibration, though executed in a straight line instead of in the arc of a circle, w^e may pro- perly call a pendulum-Vibration, as expressing the law according to which it takes place. This law admits of easy geometrical illustration as follows. Let a ball, or other small object, be attached to some part of an upright wheel revolving uniformly about a fixed axis, so that the ball goes round and round in the 38 ZA W OF PEXD VL UJI-YIBRA TIOX. [I. § 20. same vertical circle with constant velocity. If the sun is in the zenith, i. e. in such a position that the shadows of all objects are thrown vertically, the shadoiv of the hall on any liorizontal 2^lctne heloiv it will "iiiove exactly as does the hob of a pendulum. The form of the associated wave for longitudinal pendulum-vibrations is show^i in Fig. 17 {a). F.g.nCa) Retaining the form of the cur^^e, we may make its amplitude and wave-length as large or as small as we please, as in the case of the waves in Fig. 4, (1) and (2), p. 13. 20. We have examined the transmission of weaves due to longitudinal vibrations along a single very slender filament. Suppose that a great number of such filaments are placed side by side in contact with each other, so as to form a uniform material column. If, now, precisely equal waves are trans- mitted along all the constituent filaments simulta- I. § 20.] ATMOSPHEPJC PRESSURE. 39 neously, successive pulses of condensation and rare- faction will pass along the column. The parts in any assigned transverse section of the column will, obviously, at any given moment of time, all have exactly the same degree of compression or dilatation. When a pulse of condensation is traversing the section, its parts will be more dense, when a pulse of rarefaction is traversing it, less dense, than they would be, were the column transmitting no waves at all, and its separate particles, therefore, absolutely at rest. Let the column with which we have been dealing be the portion of atmospheric air enclosed within a tube of uniform bore. The phenomena just described will then be exactly those which accompany the passage of a sound from one end of the tube to the other. It remains to examine the mechanical cause to which these phenomena are due. Atmospheric air, in its ordinary condition, exerts a certain pressure on all objects in contact with it. This pressure is adequate to support a vertical column of mercury 30 inches high, as we know by the common barometer. In Fig. 18 is shown a sec- tion of a tube closed at one end, with a moveable piston fitting into the other. In (1) the air on both sides of the piston is in the ordinary atmospheric condition, so that the pressure on the right face of 40 2IARI0TTFS LAW. [I. § 20. the piston is counteracted by an exactly equal and opposite pressure on its left face. (0 F^.ia. (^) (3) In (2) the piston has been moved inwards, so as to compress the air on the right of it. That on its left, being in free communication with the external air, is not permanently affected by the motion of the piston. In order to retain the piston in its forward position, it is necessary to exert a force upon it, in the direction of the arrow. If this force is relaxed, the piston is driven back. Since the pressure of the air on the left of the piston is just what it was before, that on its right must necessarily have in- creased. But this increase of pressure is accom- panied by an increase of density, due to the com- pression of the air on the right of the piston. Hence increase of pressure accompanies increase of density. I. § 21.] MARIOTTE'S LA W. 41 If, as in (3), we reverse tlie process, by moving the piston outwards, the extraneous force must be ex- erted in the opposite direction, as shown by the arrow. The pressure on the right of the piston is therefore less than the normal atmospheric pressure on its left, i.e. diminution of pressure accompanies diminution of density. By experiments such as the above, it was shown, by the French philosopher Mariotte, that the pressure of air varies as its density. 21. Next, let us take a cylindrical tube open at one end and having a moveable piston fitting into the other, as in Fig. 19. In (1) the piston is at rest at A, and the air in its ordinary atmospheric condition of density and Ti^.19 0) (*) — ^1 1 A S = ^11 1 c D II 1 1 J' E F o ^1 1 1 1 H K L M 11 1 1 1 i IJ pressure. In (2) the piston is pushed inwards as far as C. AVhile it is moving up to this position, the 42 TRAXS2nSSI0X OF PULSES. [I. § 21. air-particles in front of it are thrown into motion. Suppose that, at the moment when the piston reaches C, the particles at D are just beginning to be dis- turbed. The air which, in (l), occupied AB, is now crowded into CD, and is, therefore, denser than that further on in the tube. Now, let the piston be drawn back to E, (3), as much to the left of its original position, ^, (l), as, in (2), it was to the right of it. The air in CD, (2), will, while this is taking place, expand into EF; for, being denser, it will also be at a greater pressure, than the air to the right of it. It will, therefore, act on the air in advance of it in the same way as the piston did on the air in contact with it when moving from ^, (1) to C, (2). Hence the air in FG will be condensed, G being the point where the air particles are just beginning to be disturbed when the piston reaches the position E. Thus the air at D advances to F. Further, in consequence of the backward motion of the piston, the air in the neighbourhood of C, (2), has to move to E, (3). Thus the air originally in AB now occupies EF, which is greater than AB. It is therefore less dense than in (l), i.e. is in a state of rarefaction. Now, let the piston again advance to II, (4). The air in FG being at a greater pressure than that in its front, and still more so than that in its rear, Avill expand in both directions, causing a I. § 21.] TBAXSMISSIOX OF SOUXD. 43 new condensation, LM, to be formed further on, and itself becoming the rarefaction KL, co-operating, at the same time, with the advancing piston to pro- duce in its own rear the condensation HK. In (5) the piston is again where it was in (3). HK has expanded into the rarefaction NO, KL contracted into the condensation OP, LM expanded into the rarefaction PQ, and a new condensation, QR, been formed in front. The fissure makes it clear that each forward stroke of the piston produces a pulse of condensation, and each backward stroke a pulse of rarefaction ; but that, when once formed, these pulses travel onwards independently of any external force. They do so, as we have seen, in virtue of the relation which connects the pressure of the air with its density, in other Avords, on the elasticity of the air. If we suppose our moveable piston withdrawn from the tube, and a vibratino- tunino'-fork held with the extremity of one prong close to the orifice of the tube, the conditions of the problem will not be essentially modified. Each outward swing of the prong will give rise to a condensed, and each inward swing to a rarefied pulse, and thus, during every com- plete vibration of the fork, one sonorous wave, con- sisting of a pulse of condensation and a pulse of rare- faction, will be started on its journey along the tube. 44 FREELY EXPANDIXG SOUXD-WAVE. [L § 22. 22. We have examined the transmission of Sound along a cokimn of air contained in a tube of uniform bore. A more important case is that in which a sound, originated at an assigned point, spreads out from it freely in all directions. Here we must conceive a series of spherical shells, alternately of condensed and of rarefied air, one inside the other, and all having the point of origination of the sound as their common centre. All the shells must be supposed to expand uniformly like an elastic globular balloon constantly inflated with more and more gas. The great difference between this case^ and that last considered lies in this, that, as the spherical shells of condensation and rarefaction spread, it is necessary, in order to keep up the wave-motion, to throw larger and larger surfaces of air into vibra- tion ; whereas within the tube the transverse section remained the same throughout. Hence, as the same amount of original disturbing force has to set a constantly increasing number of air-particles into motion, it can only do so by proportionately shorten- ing the distances through which the individual particles move, i. e. by dimmishing their extent of vibration. Accordingly when Sound-waves spread out freely in all directions, the further any given air-particle is from the point at which the sound originated, the smaller will be the extent of the I. § 23.] MUSICAL AND NON-MUSICAL SOUNDS. 45 vibration into wliicli it will be thrown when the waves reach it. 23. Sounds are either musical or non-musical. The vast majority of those ordinarily heard— the roaring of the wind, the din of traffic in a crowded thoroughfare — belong to the second class. Musical sounds are, for the most ]3art, to be heard only from instruments constructed to produce them. The dif- ference between the sensations caused in our ears by these two classes cf sounds is extremely well marked, and its nature admits of easy analysis. Let a note be struck and held down on the harmonium, or on any instrument capable of producing a sustained tone. However attentively we may listen, we per- ceive no change or variation in the sound we hear. A perfectly continuous and imiform sensation is experienced as long as the note is held down. If, instead of the harmonium, we employ the pianoforte, where the sound is loudest directly after the moment of percussion, and then gradually dies away, the result of the experiment is that the diminution of loudness is the only change which occurs : the effect produced is the same as if our harmonium had, while sounding out its note, been carried gradually further and further away from us. In the case of non-musical sounds, variations of a different kind can be easily detected. In the howl- 46 STEADY AND UNSTEADY SOUNDS. [I. § 23. ino- of the wind the sound rises to a considerable degree of shrilhiess, then falls, then rises again, and so on. On parts of the coast, where a shingly beach of considerable extent slopes down to the sea, a sound is heard in stormy weather which varies from the deep thundering roar of the great breakers, to the shrill tearing scream of the shingle dragged along by the retreating surf. Similar variations may be no- ticed in sounds of small intensity, such as the rust- ling of leaves, the chirping of insects, and the like. The difference, then, between musical and non- musical sounds seems to lie in this, that the former are constant, while the latter are continually varying. The human voice can produce sounds of both classes. In singing a sustained note it remains quite steady, neither rising nor falling. Its conversational tone, on the other hand, is perpetually varying in height even withm a single syllable; dbectly it ceases so to vary, its non-musical character disappears, and it becomes what is commonly called 'sing-song.' We may then define a musical sound as a steachj sound, a non-musical sound as an unsteachj sound. It is true we may often be puzzled to say whether a particular sound is musical or not : this arises, however, from no defect in our definition, but from the fact that such sounds consist of two elements, a musical and a non-musical, of which the latter may I. § 23.] STEADINESS OF MUSICAL SOUNDS, 47 be the more powerful, and therefore absorb our at- tention, until it is specially directed to the former. For instance, a beginner on the violin often produces a sound in which the irregular scratching of the bow predominates over * the regular tone of the string. In bad flute playing, an unsteady hissing sound accompanies the naturally sweet tone of the instru- ment, and may easily surpass it in intensity. In the tones of the more imperfect musical instruments, such as drums and cymbals, the non-musical element is very prominent, while in such sounds as the hammering of metals, or the roar of a water-fall, we may be able to recognize only a trace of the musical element, all but extinguished by its boisterous com- panion. We have seen that Sound reaches our ears by means of rapid vibrations of the particles of the atmosphere. It has also been shown that steadiness is the characteristic feature of musical, as distin- guished from non-musical, sounds. We may infer hence that the motion of the air corresponding to a single musical sound wull be itself steady, i. e. that equal numhers of equal vibrations luill he executed in precisely equal times. This conception of the physical conditions under which musical sounds are produced will suffice for the present. We proceed to consider in detail the various ways in which such 48 STEADY AIR-MOTION. [I. § 23. sounds may differ from each other, and to investi- gate the mechanical cause to which each such dif- ference is to be referred. In what follows, by the word ' sound ' will always be meant ' musical sound,' unless the contrary be expressly stated. CHAPTER II. ON LOUDNESS AND PITCH. 24. A musical sound may vary in three difierent respects. Let a note be played, first by a single violin, then, by two, by three, and so on, until we have all the violins of an orchestra in unison upon it. This is a variation of loudness only. Next, let a succession of notes be played on any instrument of uniform power, such as the harmonium without the expression-stop, or on the principal manual of an organ, only one combination of stops being in either case used. Here we have a variation of jpitch alone. Lastly, let one and the same note be suc- cessively struck on a number of pianofortes of the same size, but by different makers. The sounds heard will all have exactly the same pitch, and ctbout the same degree of loudness ; nevertheless they will exhibit decided differences of character. The tone of one instrument will be rich and full, of another ringing and metallic, that of a third will be described as ' wiry,' of a fourth as ' tinkling,' and so on. Sounds thus related to each other are said to vary in quality only. The instances just considered T. 4 50 ELE2IEXTS OF A MUSICAL SOUXD. [II. § 25. Lave tlie advantage of simplicity, since tliey allow of changes in loudness, jDitcli, and quality being exhibited separately. They are, however, less strik- ing than other cases where sounds vary in two, or in all three, of these respects at the same time. A practised ear may be requisite to detect the difference between the tone of two pianofortes, but no one is in dano'er of mistakmo- for instance, a flute for a trumpet. There is here, no doubt, considerable differ- ence of loudness as well as of quality, but let the more powerful instrument be placed at such a dis- tance that it sounds* no louder than the weaker one, and the distinction between the tvv'o kmds of tone will be still quite decisive. Two assigned musical sounds thus may differ from each other in loudness or pitch or quality, and agree in the other two — or they may differ in any two of these, and agree in the thu-d — or they may differ in all three. There is, however, no other respect in which they can differ, and accordingly we know all about a musical sound as soon as we know its loudness, its pitch, and its quality. These three elements determine the sound, just as the lengths of the three sides of a triangle determine the triangle. 25. The loudness of a musical sound depends entirely, as we shall easily see, on the extent of oscillatory movement performed by the individual II. § 25.] LOUDNESS AND EXTENT OF VIBRATION. 51 particles composing tlie medium througli wliicli tlie sound is conveyed to our ears. A sound-producing instrument can be readily observed to be in a state of rapid vibratory motion. The vibrations of a tuning-fork are perceptible to the eye in the fuzzy, half-transparent, rim which surrounds its prongs when it is struck; and to the touch, if, after striking the fork, we place a finger gently against one of the prongs. The harder we hit the fork the louder is its sound, and the larger, estimated by both the above modes of observation, are its vibrations. The experiment may be tried equally well on any piano- forte whose construction allows the wires to be uncovered. It is natural to infer that a vibration on the part of a sound-producing instrument com- municates to the particles of the air in contact with it a corresponding movement. Thus a sound of given loudness is conveyed by vibrations of given extent, and, if the sound increases or diminishes in intensity, the extent of the vibrations will increase or diminish with it. We conclude, then, that the loudness of a musical sound depends solely on the extent of excursion of the particles which constitute the conveying medium m the neiglibourliood of our ears. This last condition is clearly essential, since a sound grows more and more feeble, the greater our distance from the point 4—2 52 FITCH AND RAPIDITY OF VIBRATION. [II. § 26. where it is produced. This diminution of intensity with the increase of distance from the origin of sound is a direct consequence of the connection be- tween loudness and extent of vibration. We have seen [§ 22] that the further an air particle is from the point where a sound is produced, the smaller will be the extent of the vibration into which it is thrown by the sonorous wave. Hence, as the sound advances, it will necessarily become feebler, provided always that the waves are permitted to spread out in all direc- tions. If they are confined, say, in a tube, the intensity of the sound will not diminish with any- thing like the same rapidity. We have here the theory of message-pipes, which are used in large establishments to enable a conversation to be carried on between distant parts of a building. A whisper, inaudible to a person close to the speaker, may, by their means, be perfectly well heard by a listener at the other end of the tube. 26. We have next to enquire to what mechani- cal causes differences in the j/^iYc/Z' of musical sounds are to be referred. Rough observation at once in- dicates the direction in which we must look. If we draw the point of a pencil along a rough surface, first slowly and then more quickly, the sound heard will be distinctly shriller the more rapid the move- ment of the pencil. As its point passes over the II. § 26.] THE SYREX. d3 minute elevations and depressions wliicli constitute the roughness of the surface, a series of irregular vibrations are set up in the materials of the surface, and by them communicated to the air. The more rapid are these vibrations, the shriller does the sound become. The instrument described below, which is called a ' Syren,' gives us the means of following up with accuracy the hint just obtained. ^ Fi9'20 AB is a thin circular disc of tin or card-board, which, by means of a multiplying wheel, can be set in rapid revolution about a fixed axis through its centre, C. A series of holes (eight in the figure) are punched in the disc at equal distances along a circle having its centre at C. A small tube, ah, is held with one end close to one of the holes. If, while the disc is rotating, we blow steadily and continuously into the tube at a, a certain quantity of air will pass through the disc whenever a hole traverses the 54 THE SYREy. [IT. § 27. orifice h, of tlie tube ah. During tlie intervals of time wHcli elapse between tlie passage of adjacent holes across h, no air can pass through the disc. Hence, if the disc be revolving uniformly, a series of such discharges will succeed each other at perfectly regular intervals of time. The air on the other .side of the disc will necessarily be agitated by the process. Every time that air is driven through one of the holes, an increase of pressure occurs close to it, and accordingly a pulse of condensation is formed there. The elastic force of the air will give rise to a pulse of rarefaction during each interval between successive discharges. Hence the Syren supplies us with a regular series of alternate condensations and rare- factions which when sufficiently raj^id will, as we have seen, produce a musical sound. 27. While air is being blown steadily into the tube, let the disc be made to revolve slowly, and then with gradually increasing rapidity. At first nothing will be audible but a series of faint intermittent throbs, due to the impact of the air driven through the tube against the successive portions of tlie disc which separate its holes. This sound may be exactly reproduced by moving the fore-finger to and fro rapidly before the lips, while blowing through them. It contributes nothing to the proper musical sound of the instrument, and is only audible in its imme-* II. § -21.] CONTINUITY OF PITCH. 55 diate neighbourliood. Presently, a cleejD musical sound begins to be heard, which, as the velocity of rotation increases, constantly rises in pitch. The acuteness of the sound thus obtainable depends solely on the speed to which we can urge the instru- ment, and is therefore limited only by the driving- power at our command. The rise of pitch in this experiment is perfectly continuous, that is to say, the sound of the Syren, in passing from a graver to a more acute note, goes through everi/ jpossihle intermediate degree of pitch. It is important that we should familiarize ourselves with this conception of the continuity of the scale of pitch, because in the instrument from which our ideas on this subject are usually obtained — the pianoforte — the pitch alters discontinuously , i.e. by a series of jumps of half a tone each, and we are thus tempted to ignore the intervening degrees of pitch, or even to suppose them non-existent. The more perfect musical in- struments, such as the human voice or the violin, are as capable as the Syren of passing through all degrees of pitch from one note to another in the way called 'po7^^amen^o' or 'slurring.' It is clear from the nature of the Syren's con- struction, that the only change which can take place during the rise of pitch is the increased number of impulses communicated to, and therefore of vibrations 56 MEASURE OF PITCH. [II. § 28. set up in, tlie external air, during any given interval of time. If, when a note of given pitch has been attained by the Syren, vre check any further increase of velocity, and cause the disc to rotate uniformly at the rate which it has just reached, no further alteration of pitch will occur, and the note will be steadily held by the instrument, so long as the uni- form rotation of its disc is kept up. Hence the num- her of aerial vibrations executed in a given time de- termines the pitch of the sound heard. 28. The Syren, besides teaching us this most unportant fact, gives us the means of determining the number of vibrations corresponding to any given note. If we know the number of rotations which the disc has performed in a given time, we have only to multiply this number by the number of holes on the disc in order to ascertain how many tube-dis- charges have occurred, and therefore how many corresponding aerial vibrations have been performed, in the period in question. The Syren is provided with a counting-apparatus which registers the num- ber of times its disc rotates per second. In order, therefore, to obtain the number of vibra- tions in a second which correspond to an assigned note, we have only to proceed as follows. Let the note be steadily sounded by some instrument of sustained power, e.g. organ or harmonium, and then II. § 29.] LIMITS OF MUSICAL SOUNDS 57 cause tlie Syren sound to mount the scale until its pitch coincides with that of the note under examina- tion. At the instant of coincidence read the figure indicated by the counting-apparatus. This, multi- plied by the number of holes in the disc, gives the number of vibrations per second required. It will be convenient, for the sake of shortness, to call the number of vibrations per second, to which any note is due, the vihration-numher of the note in question. It is clear, from what has gone before, that any assigned degree of pitch can be permanently re- gistered, when once its vibration-number has been ascertained. 29. The Syren shows that, below a certain rate of vibration, no musical sounds are produced. The position of the absolute limit thus placed to the gravity of such sounds cannot be exactly defined, and probably varies somewhat for different ears. The lowest note on the largest modern organs has IGJ for its vibration-number, but it is a moot ques- tion whether the musical character of this note can be recognized or not. In any case we may regard the lower limit of musical sounds as situated in the immediate neigh- bourhood of this degree of pitch. For some distance above the limit the musical character continues very imperfect, and it is not until we reach 41i vibrations 58 LIMITS OF 2IUSICAL SOUXDS. [II. § 20. per second, the lowest note of tlie double-bass, tliat we get a satisfactory musical sound. There is no corresponding limit absolutely barring the scale of pitch in the opposite direction, but sounds above a certain degree of acuteness become painful to the ear, and therefore unfit for musical purposes. The highest note of the piccolo, the shrillest sound heard in the orchestra, makes 4752 vibrations per second. And this we may regard as constituting a practical superior limit to the scale of pitch at the disposal of musical art. The extremest range attainable by exceptional human voices, from the deepest note of a bass to the highest of a soprano, lies, roughly speak- ing, between 50 and 1500 vibrations per second. Ordinary chorus voices range from 100 to 900, or 1000 vibrations per second. The number of sounds within the limits of the musical scale, which can be recognized as possessing distinct degrees of pitch, will vary with the acuteness of perception of in- dividual observers. Trained violinists are said to be able to distinguish about seven hundred sounds in a single octave, which would give nearly ^yq thousand for the whole scale. But, since the difficulty of fix- ing the pitch with accuracy increases very rapidly with very low or very high sounds, this estimate would probably much exceed the limits of what could be achieved by the very finest ear. We shall, 11. § 30.] BELATIVE PITCH. 59 however, be well within the mark if we assume that an ordinary ear can recognize, on the average, between one and two hundred sounds in an octave, or fully one thousand in the whole scale. There is nothino- in the continuously shading- off gradations of pitch to indicate what sounds should be picked out to form agreeable sequences, or combinations, with each other. Nevertheless the human mind, working on this seeming chaos from the earliest dawn of musical art, has reduced it to order by discovering the follow- ing principle. 30. When one sound has been arbitrarily selected as the starting-point, there are a certain number of other sounds, having fixed relations of pitch to that previously chosen, which are capable of forming, with it and with each other, melodic and harmonic effects especially pleasing to the ear. These are the notes of the ordinary major and minor scales, the original sound of reference being the common tonic, or keif- note, of those scales. In saying that these sounds have fixed mutual relations of pitch, we merely state formally an obvious fact. A familiar melody is recognized equally well whether heard in the deep tones of a man's, or in the shrill notes of a child's voice. Whether the singer pitches it on a low or on a hiofh note of his voice makes no difference in the melody itself In fact the correctness with which an 60 INTERVALS. [11. § 31. air is sung no more depends on tlie exact pitcli of the note on wliicli the singer starts it, than does the faithfuhiess of a plan on the precise scale which the draughtsman has adopted. It is sufficient that the constituent notes of the melody should have fixed mutual relations of pitch, just as, in the plan, the several objects represented need only be drawn in proportion to their actual dimensions. The difference in pitch of any two notes is called the interval between them : it is on accuracy of intervals that music essentially depends. 31. The most important interval in the scale is the octave. It is that which separates the highest note of a peal of eight bells from the lowest. When a bass and a treble voice sing the same melody together, the notes of the latter are usually one octave above those of the former. The octave has this peculiarity, shared by no other interval, that, if starting from any note we choose, we ascend to that an octave above it, then to that an octave above the last, and so on, we get a number of notes which sound perfectly smooth and agreeable when heard all together. The same thing holds good, if we descend by a succession of octaves from the note fixed on as our starting-point. Hence we may conveniently regard the whole scale of pitch as divided into a series of octaves, taken upwards and downwards from 11. § 31.] INTERVALS. Gl some one sound arbitrarily selected. Narrower in- tervals situated in any one octave are repeated in all the other octaves, so that, when we have settled those intervals for a single octave, we have settled them for all the rest. Within the limits of each octave, the common major scale presents us with seven notes, or, if we include that which forms the starting point of the next octave, with eight. The fact that the eighth note is the octave of the first explains the meaning of the word 'octave,' i.e. * eighth' [Latin: 'octaviis'). The eight notes are those of an ordinary peal of the same number of bells, or of the white keys of the pianoforte between two adjacent C's. We may, for convenience of reference, number them 1, 2, 3 8, beginning with the lowest note, or tonic. The fol- lowing nomenclature is used to describe the intervals formed by the several notes luith the tonic. Notes forming interyal. Name of interval. 1 and 2 Second 1 3 Major Third 1 4 Fourth 1 5 Fifth 1 6 Major Sixth 1 7 Major Seventh 1 8 Eighth or Octave. 62 COXCORDS AND DISCORDS. [II. § 32. When two notes of the same pitch are sounded to- gether, e. g. b J two histruments, or by two voices, the notes are said to be in unison. Though there is here no difference of pitch whatever, it is convenient to rank the unison as an interval. With this explana- tion we may add to the above table that 1 and 1 form the interval of an unison. The reader must carefully avoid giving to the ' Thirds,' ' Fourths,' &c., which he meets with m music the meanings attached to the same w^ords in fractional arithmetic, with which they have absolutely nothing to do. A ' Fifth,' for example, does not stand for a ffth part of an octave, or indeed for a ffth part of anything, but for the difference of pitch between the first and fifth notes of the scale. The several pairs of notes forming the intervals laid down in our table do not all produce smooth and agreeable effects when sounded together. The following pairs blend pleasantly : 1—3, 1—4, 1—5, 1—6, 1—8 ; the remaining two, 1—2 and 1—7, give rise to decidedly harsh effects. The intervals in the first line are therefore classed as concords , those in the second as discords, 32. The minor scale has the notes 1, 2, 4, 5, 8 in common with the major scale. It substitutes for 3 II. § 33.] MAJOR AND IIINOR IXTERTALS. 63 a sound lying between that note and 2, wliicli forms with 1 a consonant interval called the Mmor Third. According to cu-cumstances it may either retain G, or i-eplace it by a sound lying betv/een that note and 5, which makes with 1 a concord called the Minor Sixth. Similarly it may employ 7, or, in the room of that note, a fresh sound situated below it, but above 6, which with 1 forms a discord called the Minor Seventh. Thus, including the octave, the two scales together give us a series of eleven notes, which, severally com- bined with the tonic, form ten distinct intervals. They are expressed in musical notation as follows : — Second. Minor Tliird. Major Tliird. Fourth. Fifth. i b:-- i ^ ^=- — * — % — * — li Minor Sixth. Major Sixth. Minor Seventh. Major Seventh Octave. ISST -^&- % :q: The reader should endeavour to familiarize him- self with these intervals, so that, when he hears the pair of notes which form any one of them successively sounded, he may at once be able to name the interval. 33. The Syren enables us to obtain simple nu- merical measures of the intervals exhibited on this page. 64 MEASURE OF INTERVALS. [II. § 33. Let a second circular row, containing sixteen holes, be punclied in its disc, and the instrument set uniformly rotating. If we now blow alternately against the 8 -hole row, and the 16 -hole row, we shall find that the sound produced at the latter is pre- cisely one octave higher in pitch than the sound produced at the former. If we increase or diminish the velocity of rotation, both sounds will, of course, rise or fall proportionately, but the interval between them will remain unaffected and equal as before, to an exact octave. The number of air-discharges cor- responding to the more acute sound is, in this case, evidently twice as large, in any given time, as the number, during the same time, for the graver sound. Accordingly we have the following result. When two sounds differ hij a single octave, the higher sound makes exactly tivice as many vibrations in any assigned time as the lower. Next let a row of 12 holes be punched in the disc of the Syren. Taking this row with the 8 -hole row, and proceeding as in the last instance, we find that the more acute sound forms a Fifth Avith the graver one. The numbers of discharges in any given time are here as 12 to 8, i.e. as 3 to 2. The result therefore is as follows : ivhen tivo sounds differ hy a Fifth, the higher sound makes exactly three vibrations during the time in ivhich the lower sound makes tivo. 11. §§ 34, 35.] MEASURE OF INTERVALS. 65 If we take the 16-liole and 12-liole rows together, the interval amounts to a Fourth ; accordingly, when two sounds differ hy a Fourth, the higher sound makes exactly four vibrations during the time inivhich the lower sound makes three. 34. The results just obtained may be somewhat more concisely stated. During one second of time, the upper of two sounds differing by an octave makes a number of vibrations, which is to the number made by the lower sound as 2 to 1. For a Fifth the ratio is as 3 to 2. For a Fourth it is as 4 to 3. Remem- bering, then, the definition of the vibration-number of an assigned sound [§ 28], we may express our three results as follows : — When two sounds form with each other the intervals of an octave, a Fifth or a Fourth, their vibration-numbers are to each other, in the first case as 2 to 1, in the second as 3 to 2, in the third as 4 to 3. A ratio is most easily expressed by a fraction. Thus we may regard the fraction | as denoting the interval of a Fifth. It may be taken as an abbre- viated statement of the fact that, when two sounds form a Fifth with each other, the more acute makes 3 vibrations while the graver makes 2. 35. By suitable experiments, similar numerical relations to those already established may be ob- T. 5 66 YIBRA TIOX-FRA CTIONS. [II. § 35. tained for all the intervals already considered. A fraction can thus be determined for each interval, in the manner exemplified in the case of the Fifth. We will call this fraction the vibration-fraction of the interval in question. The accompanying table gives, in the second column, the vibration-fractions corre- sponding to the intervals named in the first ; and in the third, describes the consonant or dissonant cha- D^acter of the intervals. Name of interval. Vibration-fraction. Character of interval. Unison 1 concord Second 9 8 discord Minor Third f ' concord Major Third 5 4 concord Fourth 1 concord Fifth 1 concord Minor Sixth S 5" concord Major Sixth 3 concord Minor Seventh 13 y discord Major Seventh 15 8 discord Octave f concord It is noticeable that the dissonant intervals in- volve higher numbers in their vibration-fractions than the consonant intervals do ; the latter, with the solitary exception of the Minor Sixth, having nothing beyond G, while the former bring in 9, 15 and 16. II. § 36.] CALCULATION OF VIBRATION-NUMBERS. 67 36. By the help of the last table, we can calcu- late the vibration-numbers of all the notes within a single octave which belong to the major or minor keys as soon as the vihration-numher of the tonic is given. For instance, let middle C of the pianoforte (vib.-no. 264) be the tonic. From the second line of the table, we see that the vibration-number of D must be to 264 in the ratio of 9 to 8. It must 9 therefore be equal to - x 264, or 297. For E[,, by /? exactly similar reasoning, we obtain - x 264 or 316f; o for E, ^ X 264, or 330. The student should work out the remaining cases for himself. The complete results for the major scale are as follow : — fe 264 297 SSO 852 896 440 In order to extend the scale another octave up- wards, we have only to multiply each vibration-num- ber by 2. A second multiplication by 2 will raise it by another octave, and so on. Conversely, in order to pass to the octave below, we divide each vibration- number by 2. To descend a second octave we repeat the operation, and so on. 5—2 G8 CALCULATION OF VIBEATIOF-NUMBERS. [11. § 36. Thus the pitch of the tonic absolutely fixes the pitch of every note, in the scale of which it is the starting-point. Before we proceed to investigate the mechanical equivalent of the thu'd element [§ 24] of a musical sound, its quality, it will be convenient briefly to examine a subject possessing an important bearing on that enquiry. This we shall do in the next chapter. CHAPTER III. ON BESONANCE. 37. When a sounding body causes another body to emit sound, we have an instance of a very remark- able phenomenon called resonance. The German term for it, ' CO -vibration' [Mitscliivingung), possesses the merit of at once indicating its essential meaning, namely, the setting up of vibrations in an instrument, not by a blow or other immediate action upon it, but indirectly as the result of the vibrations of another instrument. In order to produce the effect, we have only to press down very gently one of the keys of a pianoforte, so as to raise the damper, without making any sound, and then sing loudly, into the instrument, the corresponding note. When the voice ceases, the instrument will continue to sustain the note, which will then gradually fade away. If the key is allowed to rise again before the sound is extinct, it will abruptly cease. A similar experiment may be tried, as follows, on any horizontal pianoforte which allows the wires to be uncovered. Each note is, it is well 70 RESONANCE, [III. § 37. known, produced by two, or by three, wires. Having, as in the previous case, raised one of the dampers without striking the note, twitch one of the corresponding wires sharply with the finger-nail, and then wait a few seconds. The vibrations will, in this interval, have communicated themselves to the other string, or strings, belonging to the note pressed down : if, now, the first wire be stopped by applying the tip of the finger to the point where it was at first twitched, the same note, produced by these trans- mitted vibrations, will continue to be sustained by the remaining wire or wires. A more instructive method of studying resonance is to take two unison tuning-forks, strike one of them, and hold it near the other, but without touch- ing it. The second fork will then commence sound- ing by resonance, and will continue to produce its note though the first fork be brought to silence. It is essential to the success of this experiment that the two forks should be rigorously in imison. If the pitch of one of them be lowered by causing a small pellet of wax to adhere to the end of one of its prongs, the effect of resonance wHl no longer be produced, even though the alteration of pitch be too small to be recognized by the ear. Further, the phenomenon re- quires a certain appreciable length of time to develope itself; for, if the silent fork be only momentarilj ex- III. § 37.] ACCUMULATED IMPULSES. 71 posed to the influence of its vocal fellow, no result ensues. The resonance, when produced, is at first extremely feeble, and gradually increases in intensity under the continued action of the originally- excited fork. Some seconds must elapse before the maximum- resonance is attained. The conditions of our experi- ment show, directly, that the resonance of the second fork was due to the transmission, hij the air, of the vibrations of the first, the successive air- impulses fall- ing in such a manner on the fork as to produce a cumulative effect. If we bear in mind the dispropor- tionate mass of the body set in motion compared to that of the air acting upon it, — steel being more than six thousand times as heavy as atmospheric air, for equal bulks, — we cannot fail to regard this as a very surprising fact. Let us examine the mechanical causes to which it is due. Suppose a heavy weight to be suspended from a fixed support by a flexible string, so as to form a pendulum of the simplest kind. In order to cause it to perform oscillations of considerable extent by the application of a number of small impulses, we proceed as follows. As soon as, by the first impulse, the weight has been set vibrating through a small distance, we take care that every succeeding impulse is impressed in the direction in luhich the iveight is moving at the time. Each impulse, thus applied, will 72 ACCUMULATED IMPULSES. [III. § 37. cause the pendulum to oscillate tlirougli a larger angle than before, and, the effects of many impulses being in this way added together, an extensive swing of the pendulum is the result. When the distance throuo-h which the weio-ht travels to and fro, though in itself considerable, is small compared to the lerajth of the supporting string, the time of oscillation is the same for any extent of swing within this limit, and depends only on the length of the strmg. My readers will find this im- portant principle illustrated in any Manual of Elemen- tary Mechanics, and I must ask them to take it for granted here. For the sake of simphcity, let us sup- pose that we are dealing with a second's pendulum, {. e. one of such a length as to perform one complete oscillation in each second, and therefore to make a single forward or backward swing in each half second. It will be clear, from what has been said above, that the most rapid effect will be produced on the motion of the pendulum, by applying a forward and a back- ward impulse respectively during each alternate half second, or, which is the same thmg, administering a pair of to and fro impulses during each complete oscillation of the pendulum. We have a simple in- stance of such a proceeding in the way in which a couple of boys set a heavily laden swing in violent motion. They stand facing each other, and each boy. III. § 38.] CA USE OF RESONANCE. 73 when the swing is moving away from him, helps it along with a vigorous push. 38. The above considerations enable us to ex- plain how a sounding-fork excites the vibrations of another fork in unison with itself, through the me- dium of the intervening air. When a continuous musical note is being sounded, we know that, at any one point we choose to fix upon, the air is undergoing a series of rapid changes, becoming alternately denser, and less dense; than it would be were the sound to cease. The increase of density is accompanied by an increase of pressure ; its diminution by a diminution of pressure [§ 20]. Tig. 21 y Let Ay Fig. 21, be the sounding-fork, B that whose vibrations are to be excited by resonance, and let us consider the effect of the alternations of pres- sure on the air at c on the prong he. The increase of pressure will tend to move the prong into the posi- tion hd, its subsequent diminution will facilitate the elastic recoil of the fork, supported also by the su- perior density of the air on the other side of the 74 CACSF OF EESONAXCE. [III. § 38. prong, and thus tend to bring the prong mto the po- sition he, further to the left of its original position, l)c, than hd was to the right of it. Thus the alter- nate condensations and rarefactions of the Sound- waves impress on the fork B corresponding impulses in opposite directions. One pair of such impulses is applied regularly during each complete vibration of B, since they are due to the vibrations of A, which is in unison with B. Further, for the small extent of vibration with which we have here to deal, the prongs of a tuning-fork move exactly according to the same law as a j)endulum^. Accordingly, these au^-impulses are applied under precisely the conditions which we found to be most favourable to the rapid develop- ment of vibratory motion. The large number of such impulses which succeed each other in a few seconds, make up for the feebleness of each by itself. It is in accordance with this, that resonance is produced more slowly between unison-forks of low, than be- tween those of high, pitch. I find that, with two making 256 vibrations per second, about one second is requisite to bring out an audible resonance ; while with another pau", making 1920 vibrations per second, I am not able to damp the first fork sufficiently soon after striking it, to prevent the other from makmg itself heard. ^ This will be proved in § 70. TIL § 39.] RESONANT AIR-COLUMNS. 75 39. A column of air is easily set in resonant vibration by a note of suitable pitch. The roughest experiment suffices to establish this fact. We have only to roll up a piece of paper, so as to make a little cylinder six inches long and an inch or two in diameter, with both ends open, and to hold a com- mon C tuning-fork |g - = nA close to one of the apertures, after striking it briskly. As soon as the fork reaches the position (l) Fig. 22, its tone will un- mistakeably swell out. In order to estimate the in- crease of intensity produced, it is a good plan to move the fork rapidly to and fro, a few times, be- tween the positions (1) and (2). Tij.22 In the first case we have the full effect of resonance, in the second only the unassisted tone of the fork, and the contrast is very marked. We may shorten or lengthen our cylinder, within certain limits, and 76 RESOXANT AIR-C0LU2fNS. [III. § 39. still obtain the phenomena of resonance, but the greatest reinforcement of tone we can attain with the fork selected will be produced by an air-column about six inches long. If we close one end of the paper cylinder, by placing it, for instance, on a table, and repeat our ex- periment at the open end, only a very weak resonance is produced ; but we obtain a powerful resonance by operating mth a fork ( W^=^^) niaking half as many vibrations per second as that before em- ployed. In this case, then, a column of air contained in a cylinder, of which one end was closed, resounded powerfully to a note one octave below that which elicited its most vigorous resonance when contained in a cylinder open at both ends. By operating in this fashion, with forks of dif- ferent pitch, on air-columns of different lengths, we arrive at the following laws, which are universally true : — 1. For every single musical note there is a cor- responding ak-column of definite length which re- sounds the most powerfully to that note. 2. The maximum resonance of air in a closed pipe is produced by a note one octave below that to which an open pipe of the same length resounds the most powerfully. III. § 40.] RESONANT AIR-COLUMNS, 77 40. In order to ascertain the precise relation between the pitch of a note and the length of the corresponding air- column, we will examine the way in which resonance is produced in a column of air contained in a pipe closed at one end. Let A, Fig. 23, be the open, and B the closed, ends of the pipe, and let us, for a moment, replace the contained air by an elastic spiral spring fastened at B, and of length equal to AB, rwmvmuumrwfuuw B Suppose the end of the spring suddenly pushed a little way from A towards B. The coils of the spring nearest A will be squeezed together, and this condensed state of the spring will travel along it until it reaches B. The end of the pipe will there cause the condensation to rebound, and travel back again to ^. If let alone, the end of the spring would now protrude slightly beyond the open end of the tube, the coils near A would be drawn some- what apart and a rarefaction would in consequence pass along AB and after reflexion at B return to A, where it would meet the end of the spring just contracting to its original length. The elasticity of the spring would, thus, cause it to lengthen and shorten as a whole, in consequence of the single push 78 RESOFANT ATR-COLUMNS. [III. § 40. originally given it, and this motion would for a time continue, its successive periods being four times the space of time occupied by a pulse of condensation or rarefaction in traversing the length of the tube. The free end of the wire may, however, be pulled and pushed, alternately, so as to reinforce each pulse as it arrives at the open end of the tube, and in this manner the maximum of motion will be com- municated to the spring. In this case, one outward, and one inward, impulse of the hand must be com- municated to the free end of the string, during the time which elapses while a pulse traverses four times the length of the tube. Eeverting to the actual conditions of our problem, we have the reso- nance of the air-column, in place of the alternate lengthening and shortening of the spring. For the to and fro motion of the hand at ^, we must substi- tute that of the prong of the vibrating fork. The sound-pulse traverses four times the length of the tube while the fork is performing one complete vibration. We know, however [§§ 8 and 15], that, during this latter period, the sound-pulse due to the fork's action traverses precisely one wave-length corresponding to the pitch of the note produced by the fork. Hence, for maximum resonance in the case of a closed pipe, the wave-length corresponding to the note sounded must be four times as great ITI. §§ 41, 42.] RESONANCE-BOXES. 79 as the lenortli of the ah^-column, or the leno;th of the cohimn one quarter of the wave-length. 41. These prmciples give us the explanation of a useful appliance for intensifying the sound of a tuning-fork. Such a fork, when held in the hand after being struck, communicates but little of its vibration to the surrounding air; when, however, its handle is screwed into one side of an empty wooden box of suitable dimensions, in the way shown in Fig. 24, the tone becomes much louder. The vibrations of the fork pass from its handle to the w^ood of the box. a:d thence to the air-column within, which is of appropriate length for maximum resonance to the fork's note. This convenient adjunct to a tuning- fork goes by the name of a 'resonance-box.' 42. When a number of musical sounds are going on at once, it is generally difficult, and often impossible, for the unaided ear to decide whether an individual note is, or is not, present in the whole mass of sound heard. If, however, we had an instrument which intensified the tone of the note of 80 RESONATORS. [III. § 42. wliich we were in search, without similarly rein- forcing others which there was any risk of our mistaking for it, our power of recognizing the note in question would be proportionately increased. Such an instrument has been invented by Helmholtz. It consists of a hollow ball of brass with two aper- tures at opposite ends of a diameter, as shown in Fig. 25. The larger aperture allows the vibrations of the external air to be communicated to that within the ball ; the smaller aperture passes through a nipple of convenient form for insertion m the ear of the observer. The air contained in the ball resounds very powerfully to one single note of definite pitch, whence the instrument has been named, by its in- ventor, a resonator. The best way of using it is, first, to stop one ear closely, and then to insert the nipple of the instrument in the other; as often as the III. § 42.] RESONATORS. 81 resonator's own note is sounded in the external air, the instrument will sing it into the ear of the ob- server with extraordinary emphasis, and thus at once single out that note from among a crowd of others differing from it in pitch. A series of such resona- tors, tuned to particular previously selected notes, constitutes an apparatus for analyzing a composite sound into the simple tones of which it is made up. T. CHAPTER lY. ON QUALITY. 43. The laws of resonance enable us to establish a remarkable, and by most persons utterly un- suspected, fact, viz. that the notes of nearly every regular musical instrument with which we are familiar, are not, as they are ordinarily taken to be, single tones of one determinate pitch, but composite sounds containmg an assemhlage of such tones. These are always members of a regular series, form- ing fixed intervals with each other, which may be thus stated : if we number the separate single tones, of which any given sound is made up, 1, 2, 3, &c., beginning with the lowest, and ascending in pitch, we have (1) The deepest, or fundamental, tone, which is commonly treated as determining the pitch of the whole sound. (2) A tone one octave above (1). (3) Atone a Fifth above (2), i.e. a Twelfth above (1). ly. § 43.] CONSTITUENTS OF COMPOSITE SOUNDS 83 (4) A tone a Fourth above (3), i.e. two octaves above (l). (5) A tone a Major Third above (4), i.e. two octaves and a Major Third above (1). (6) A tone a Minor Third above (5), i.e. two octaves and a Fifth above (1). These are the most important members of the series. Their vibration-numbers are connected by a simple law, which is easily deduced from the above relations. If the fundamental tone makes 100 vibrations per second, (2) will make twice as many i.e. 200 ; (3) being a Fifth above (2), will have for 3 its vibration-number, - x 200, or 300. For (4), which is a Fourth above (3), we get similarly ^-x 300, or 400; for (5) - x 400, or 500; for (6), - X 500, or 600. Thus the numbers come out 100, 5 200, 300 and so on, or, generally, whatever be the vibration-number of (1), those of (2), (3), (4), &c., are respectively twice, three times, four times, do. as great. Subjoined, in musical notation, is the series of tones complete up to the tenth, taking C in the bass clef as our fundamental tone, though any other would do equally well. 6—2 84 . SEPdES OF COXSTITUEXT TONES'. [IV. § 44. ^^. .5. ^ -e i^=g: _<2. The asterisk denotes that the pitch of the 7 th tone is not precisely that of the note by which it is here represented. It is in fact sHghtly less acute. The reader must not suppose, that, because the tones into which a note of a musical instrument may usually be decomposed are members of a fixed series, all those which we have written down are neces- sarily present in every such note. All that is meant to be asserted is, that those which are present, be they few or many, must occupy positions deteraiined by the law connectmg each tone with its funda- mental. The sound may contain, say, (l), (3), and (5) only, or (l), (4) and (8) only, and so on, the rest being entirely absent, but 171 no case can a tone intermediate in 'pitcli between any two consecutive members of the series make its appearance. 44. Experimental evidence shall now be pro- duced in support of the extremely important pro- position just enunciated. We will begin with the sounds of the pianoforte. Let the note W-^hztz be first silently pressed down, IV. § 44.] AXALYSIS BY RESOXAKCE. 85 and tlien ^~ n~z be vigorously struck, and, after three or four seconds, allowed to rise ao-ain. The lower note is at once extinguished, but, we now hear its octave sounding with considerable force from the wires of 5^— — . If we jDermit the damper to fall back on these, by releasing the note hitherto held down, the whole sound is immediately cut off. Next, retaining the same fundamental note, ^—a — , let ^ tE^= be quietly freed from its damper, and the experiment repeated as before. We shall then hear this note sounding on after the extinction of ^^^—n — . Similar- results may be obtained with the three next tones, gg=^E==, but they drop off very rapidly in intensity. The tones above ^ are so weak as to be practically insensible. The series of tones produced in this succession of trials can only be due to resonance. But, as has been already shown, the vibrations of any instrument are excited, by resonance, onhj when vibrations of the same 2)eriod are already present in the surrounding air. Accordingly, the only sound directly origmated in each variation of our experiment, viz. that of the note 5^gy — , must have contained cdl the tones sue- 86 AHIALYSIS BY RESONANCE. [IV. § 45. cessively heard. The reader should apply the method of proof here adopted to notes in various regions of the key-board. He will find considerable differences, even between consecutive notes, in the number and relative intensities of the separate tones into which he is thus able to resolve them. The higher the pitch of the fundamental tone, the fewer will the recognizable associated tones become, until, in the region above the notes are themselves approximately single tones. The causes of these differences will be explained, in detail, in a subsequent chapter ; it is sufficient here to indicate their existence. The result arrived at, thus far, is that the sounds of the piano- forte are, in general, composite, the number of consti- tuent tones into which they are resolvable being- largest in the lower half of the instrument, and dimi- nishing in its upper half, until, at last, no analysis is called for. 45. The above resolution has been effected by means of the principle of resonance. It can, however, be performed by the ear directly, though only to a small extent, and with less ease. In endeavouring to hear a particular constituent tone among the as- semblage forming a compound sound, the best plan is, f rst to let the upper tone be heard by itself a few times so as to prepare the ear for the precise degree IV. § 45.] DIRECT AXALTSJi^. 87 of pitcli it is to expect, and tlien to develope tlie compound sound. If, meanwhile, the observer has succeeded in keeping his attention unswervingly fixed on the tone for which he is listening, he will hear it come out clearly from the mass of tones included in the composite sound. If the pianoforte note, ^ q — , he thus examined, the octave, A , and Twelfth, 2 jE^z J can generally be recognised with consider- able ease ; the second octave, 2 y-^ — , with a little trouble; the next three tones of the series on p. 84, with increasing difficulty, and those which succeed them not at all. The reader approaching this pheno- menon for the first time must not be disappointed if, in trying this experiment, he fail to hear the tones he is told to expect. He should vary its conditions by changing the note struck, in such a way that his attention will not be hable to be diverted by the presence of distinct tones more acute than that of which he is in search. Thus a note near ^ may be advantageously chosen to observe the first octave. I % ^^ —- ', one near 5^— p— to observe the Twelfth, -& — i one near — to observe the second 88 DIFFICULTY OF DIRECT AXALTSIS. [lY. § 46. -- octave, i^^=: . He may however altogether fail in performing the analysis with the unassisted ear. This by no means indicates any aural defect, as he may at first be inclined to imagine. It rather shows that the life-long habit of regarding the notes of in- dividual sound-producing instruments as single tones cannot be unlearned all at once. The case is analo- gous to that of single vision with two eyes, where tioo distinct and different images are so blended together as to appear, to all ordinary observation, as one. The acoustical observer who is thus situated, must rely on the analysis by resonance, and on the evidence of those who are able to perform the direct analysis. As he pursues the subject further experimentally, his analyti- cal faculty will no doubt in time adequately develope itself 46. The composite character of musical sounds, which we have recognized in the case of the piano- forte, and shall have ample opportunity of verifying more generally in the sequel, requires the introduc- tion, here, of certain verbal definitions and limitations. The phraseology hitherto employed, both in the science of acoustics and in the theory of music, goes on the supposition that the sounds of individual instru- ments are single tones, and therefore, of course, con- tains no term specially denoting compound sounds TV. § 47.] THEORY OF QUALITY. 89 and their constituents. * Sound,' 'note/ and 'tone' are used as nearly synonymous. It will be conve- nient to restrict the meaning of the latter so that it shall denote a sound which does not admit of resolu- tion into simple elements. A single sound of deter- minate pitch we shall, accordingly, in what follows, call a tone, or simple tone. For a compound sound the word clang will be a serviceable term. The series of elementary sounds into which a clang can be re- solved we shall call its partial-tones, sometimes dis- tinguishing, among these,*the lowest, oy fundamental tone, from the others, or overtones of the clang. This nomenclature is a direct adaptation of the German terms employed by Helmholtz. Its introduction is due to Professor Tyndall. 47. This long discussion has paved the way for the complete explanation of musical quality which is contained in the following proposition. The quality of a clang depends on the numher, orders, and rela- tive intensities, of the partial-tones into which it can he resolved. We have here three different causes to which variations in the quality of composite sounds are assigned. 1. A clang may contain only two or three, or it may contain half-a-dozen, or even as many as fifteen or twenty, well-developed partial-tones. 2. The number of partial-tones present remain- 90 THEORY OF QUALITY. [lY. § 47. ing the same, the quality will vary according to the positions they occupy in the fixed series on p. 84, i.e. on their orders. Thus, a clang containing three tones may consist of (1), (2), (3), or of (1), (3), (5), or of (1), (7), (10), and so on, the quality varying in each in- stance. 3. The numher and orders of the partial-tones present remaining the same, the quality will vary according^ to the relative deofrees of loudness with which those tones speak. Thus, in the simplest case of a clang consisting of (l) and (2), (2) may be twice as loud, or as loud, or half as loud, as (l), and so on. It is clear that these three classes of variations are entirely independent of each other, that is to say, any two clangs may differ in the number, orders, and relative intensities, of their constituent partial-tones. The variety of quality thus provided for is almost in- definitely great. In order to form some idea of its extent, let us see how many clangs of different qua- lity, but of the same pitch, can be formed with the first six partial-tones, by variations of numher and. order only. We will indicate each group of tones by the corresponding figures inclosed in a bracket ; thus e. g. (1, 3, 5) represents a clang consisting of the first, third and fifth tones. All the possible groups, each necessarily contain- lY. § 47.] THEORY OF QUALITY. 91 ing tlie same fundamental tone, are given in the fol- lowino^ enumeration. Two at a time : (1, 2), (1, 3), (1, 4), (1, 5), (1, 6). Total 5. Three at a time : (1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 3, 4), (1, 3, 5), (1, 3, 6), (1, 4, 5), (1, 4, G), (1, 5, 6). Total 10. Four at a time : (1,2,3, 4), (1,2, 3, 5), (1,2,3,6), (1, 2, 4, 5), (1, 2, 4, 6), (1, 2, 5, 6), (1, 3, 4, 5), (1, 3, 4, 6), (1, 3, 5, 6), (1, 4, 5, G). Total 10. Five at a time : (1, 2, 3, 4, 5), (1, 2, 3, 4, G), (l, 2, 3, 5, G), (1, 2, 4, 5, G), (1, 3, 4, 5, G). Total 5. Six at a time : (1, 2, 3, 4, 5, G). Total 1. The whole number of groups is 31, or, if we allow the fundamental tone (l) to count by itself as a sound of separate quality, 32. Let us next ex- amine how many clangs of different quality can be obtained from a single combination of three fixed partial-tones by variations of intensity only, sup- 92 THEORY OF QUALITY. [lY. § 47.. posing tliat each tone is capable of but two degrees of loudness. Eepresenting one of tbese by /, and the other hj p, we indicate, e.g.^ by (/, p, p) a clang in which the fundamental tone is sounded forte, and the two overtones p>iano. The different cases which present themselves are the following : (///). if^PJh (PJJ), {p,P,f), ifJ^P)^ ifP^P)^ {p^f>P)> ip^P^l^) or seven in all, since (j:>, ^9, ^9) has the same quality as (//,/). The number of cases increases very rapidly as we take more partial-tones together. Thus a clang of four tones will produce 15 sounds of different quality; one of five tones 31 ; one of six tones 63, by variations of intensity only. Alto- gether we could form, with six partial-tones, each susceptible of only two different degrees of intensity, upwards of foiu^ hundred clangs of distinct quality, all having the same fundamental tone. The suppo- sition above made utterly understates, however, the varieties of quality dependent only on changes of relative intensity. A very slight increase, or diminu- tion, of loudness, on the part of a single constituent tone, is enough to produce a sensible change of quality m the clang. We should be still far below the mark if we allowed each partial tone four different degrees of intensity, though even this sup- IV. § 47,] THEORY OF QUALITY. 93 position would bring us more tlian eight thousand . separate cases. Since many more variations of inten- sity are practically efficacious, and also since the num- ber of disposable partial-tones need by no means be limited, as has here been done, to the first six, ' the above calculation will probably suffice to convince the reader that the varieties of quality which the theory we are engaged upon is capable of accounting for, are almost indefinitely numerous. This is, in fact, no more than we have a right to expect from the theory, when we reflect on the fine shades of quality which the ear is able to distinguish. No two instru- ments of the same class are exactly alike in this respect. For instance, grand pianofortes by Broad- wood and by Erard exhibit unmistakeable diflPerences, which we describe as *Broadwood tone ' and * Erard tone.' Less marked, but still perfectly recognizable, differences exist between individual instruments of the same class and maker, and even between con- secutive notes of the same instrument. To these we have to add the variations in quality due to the manner in which the performer handles his instrument. Even on the pianoforte the kinds of tone elicited by a dull slamming touch, and by a lively elastic one, are clearly distinguishable. With other instruments the distinctions are much more marked. On the violin we perceive endless grada- 94 THEORY OF QUALITY. [lY. § 4 tions of quality, from the rasj)ing scrape of the beginner up to the smooth and superb tone of a Joachim (or, as I ought rather to say, the Joachim). A precisely similar remark applies to wind instru- ments; the differences, for example, between first- rate and inferior playing on the hautbois, bassoon, horn, or trumpet, being perfectly obvious to every musical ear. In the next chapter we will discuss the quality and essential mechanism of the principal musical in- struments, among which the pianoforte will receive an amount of attention proportionate to its popu- larity and general use. We begin with the elemen- tary tones of which all composite sounds are made up. CHAPTER V. ON THE ESSENTIAL MECHANISM OF THE PEINCIPAL MUSICAL INSTKUMENTS, CONSIDEEED IN KEFERENCE TO QUALITY. 1. Sounds of tuning-forJcs. 48. When a vibrating tuning-fork is held to the ear, we perceive, beside the proper note of the fork, a shrill, ringing, and usually rather discordant, sound. If however the fork is mounted on its resonance-box, as in Fig. 24, p. 79, the fundamental tone is so much strengthened that the other is by comparison faint, and the sound heard may be regarded as practically a simple tone. It is charac- terised by extreme mildness, without a trace of anything which could be called harsh or piercing. As compared with a pianoforte note of the same pitch, the fork-tone is wanting in richness and vivacity, and produces an impression of greater depth, so that one is at first incHned to think the pianoforte note corresponding to it must be an octave lower than is actually the case. It follows 96 SIMPLE TONES. [V. § 49. immediately from the general theory of the nature of quality, that simple tones can differ only in pitch and intensity. Accordingly, we find that tuning- forks of the same pitch, mounted on resonance -boxes and set vibrating by a resined fiddle-bow, exhibit, however various their forms and sizes, differences of loudness only. When made to sound wdth equal intensity by suitable bowing, their tones are abso- lutely undistinguishable from each other. 2. Sounds of vibrating strings. 49. Sounding strings vibrate so rapidly that their movements cannot be followed directly by the eye. It will be w^ell, therefore, that we should examine how the slower and more easily controllable vibrations of non-sounding strings are performed, before treating the proper subject of this section. Take a flexible caoutchouc tube, ten or fifteen feet long, and fasten its ends to two fixed objects, so that the tube is loosely stretched between them. The tube can be set in regular vibration by impress- ing a swaying movement upon it with the fingers near one extremity, in suitable time. According to the rapidity of the motion thus communicated, the tube will take up different forms of vibration. The simplest of these is shown in Fig. 26. A and B being its fixed extremities, the tube vibrates as a FORMS OF VIBRATION. 97 whole, between the two extreme positions AaB and AbB, Tiff.SG Tlie tube may also vibrate in the form shown in Fig. 27, where AabB and AcclB are its extreme positions. In this instance the middle point of the tube, C, remains at rest, the loops on either side of it moving independently, as though the tube were fastened at (7, as well as at A and B, For this reason the point C is called a node, from the Latin nodus, a knot. Fig. 28 shows a form of vibration with two nodes. Tiff.ZB. at C and D, dividing the distance AB into three equal parts. The portions of the tube AC, CD, DB vibrate independently of each other, forming what are called ventrcd segments. We may also obtain forms with three, four, five, &c., nodes, dividing the T. 7 98 DIRECT AND REFLECTED PULSES. [Y. § 50. tube into four, ^y^, six, &c., equal ventral segments, respectively. The stiffness of very short portions of the tube alone imposes a limit on the subdividing process. Let us examine the mechanical causes to which these effects are due. 50. If we unfasten one end of the tube, and, hold- ing it in the hand as in Fig. 29, raise a hump upon it, by moving the hand suddenly through a small ^t] distance, the hump will run along the tube until it reaches its fixed extremity B ; it will then be reflected and run back to A, where it will undergo a second reflection, and so on. At each reflection the hump ivill have its convexity reversed. Thus, if while tra- velling from A towards B its form was that of a. Fig. 30, on its return it will have the form h. After reflection at -4, it will resume its first form a, and so on. Now, instead of a single jerk, let the hand hold- ing the free end execute a series of equal continuous Y. §50.] EFFECT OF OPPOSITE EQUAL PULSES. 99 vibrations. Eacli complete vibration will cause a wave ah Fig. 31, consisting of crest 6, and trough a, to pass along the tube from A to B, where reflection will turn crest into trough and trough into crest; so that the wave will return from B to A stern foremost. Next let the tube be again fastened at both ends, as before, and the vibrations of the hand impressed at some intermediate point, as C, Fig. 32. "Exj. 22.. i bi < < ^ C a gB» > ■ ' !qH 1 Two sets of waves will now starb from C in the directions of the arrows. They will be reflected at A and B, and then their eflects intermingled. We will suppose that the tube has been set in steady motion, and, the hand being removed, continues its vibrations without any external force acting on it. Two sets of equal waves are now moving with equal velocities from A towards B and from B towards A, and we have to determine their joint eflect in fixing the form of vibration in which the tube swings. Suppose that a crest a, Fig. 33, movuig from A towards B, meets an equal trough h, moving from B towards A, at the point c. The point c is now 7—2 100 FORMATION OF NODES. [T. § 51. solicited by a and h in opposite directions and witti equal energy, and therefore remains at rest. The two opposite pulses then proceed to cross each other, but, as a moves to the right and h to the left with equal speed, there is nothing to give either of them an influence upon the point c, where they first met, superior to that exercised in the contrary direction by the other. Thus c Q^emains at rest under their jomt influence, and a node is therefore formed at that 2^01 nt. If a trough had been moving from A towards B, and an equal crest from B towards A, the effect would clearly have been the same. A node must therefore he formed at every point luhere tivo equal and opposite pulses^ a crest and a trough, meet each other, 51. The annexed figure represents two series of equal waves advancing in opposite dkections with equal velocities. The moment chosen is that at which crest coincides with crest and trouo^h with trough. The joint effect thus produced does not ap- pear in the figure, our object' at present being merely to determine the number and positions of the result- ing: nodes. For the sake of clearness, one set of waves Y. § 51.] FORMATION OF NODES. 101 is represented slightly below tlie other, though, in fax3t, the two are strictly coincident. ;pid.8^. ^^4^^^^^' x-^^ Let the waves ahdf.,,z be moving from left to right, the waves zfs'n\.,a from right to left. The crest Mm meets the trough 2^nm at m. After these have crossed each other, the trough gJik and the crest rq'p will also meet at m, since hn and p?7i are equal distances. Similarly the crest efg and the trough ts'i' will meet at m. Accordingly the point m is a node, and, by exactly the same reasoning, so are a, c, ^'j 9, h Pi ^% ^j <^c. The distances between pairs of consecutive nodes are all equal, each being a single pulse-length, i. e. half a wave-length, of either series. Two pulse-lengths, as gh and ha, give three nodes g, h, and m ; three pulse-lengths four nodes, and so on. There is thus always one node in excess of the num- ber of pulses. On the other hand, the fixed ends of the tube, which are the origins of the systems of re- flected waves, occupy two of these nodes. Deducting them we arrive at this result. The number of nodes is one less than the number of the pidse-lengths (or half wave-lengths), which together make up the length of the vibrating tube. 102 NATURE OF SEGMENTAL YIBRATIOX. [Y. § 52. 52. We will now ascertain liow the portions of the tube between consecutive nodes move under the Tig. 3S. (0 .(.) (0 CO action of the two systems of waves passing along it. Let AB, Fig. 35, be the fixed ends, as before, and let us take five nodes at the points 1, 2, 3, 4, 5. In (1), the systems of Avaves coincide, accordingly each point of the tube is displaced through twice as great a distance as if it had been acted on by only one system. The tube thus takes the form indicated by the strong line in the figure. In (2), one set of waves has moved half a pulse-length to the right, and the other the same distance to the V. § 53.] RATE OF SEGMENTAL VIBRATION. 103 left. The two systems are now in complete oppo- sition at every point, and the tube is, therefore, momentarily in its undisturbed position. In (3), each system has moved through a pulse-length, and the combined effect is again produced on the tube, but in the o|)posite direction to that of (1). In (4), where the systems have moved through a pulse-length and a half, the tube passes again through its undisturbed position, and, in (5), regains the position it occupied in (1), the systems of waves, meanwhile, having each traversed two pulse-lengths, or one wave-length. Thus the tube executes one complete vibration in the time occupied by a pulse in passing along a length of the tube equal to tivice one of its oivn ventral segme^its. In other words, the tube's rate of vibration varies as the number of seg- ments into tvhich it is divided. It moves most slowly in the form shown in Fig. 26 with but a single seg- ment ; twice as fast in that of Fig. 27, when divided into two segments ; three times as fast with three segments, and so on. It is easy to confirm this by direct ' experiment, the swaying movement of the hand on the tube needing to be twice as rapid for a form of vibration with two segments as for a form with one, and so on. 53. Instead of comparing the different rates at which the same tube vibrates, when divided into 104 RATE OF SEGMENTAL VIBRATION. [V. § 53. different numbers of ventral segments, we may com- pare the rates of vibration of tubes of different lengths, divided into the same number of segments. Let us take as an example the two tubes AB, CD, Fig. 36, each divided by three nodes into four A» \ i V 3 -(D ventral segments. By what has been already shown, the time of vibration of either tube will be that which a pulse occupies in traversing two of its ven- tral segments. Therefore the time of vibration of AB will be to that of CB as Al is to (72, i. e. as one half of AB is to one half of CD, or as AB is to CD. This reasoning is equally applicable to any other case. Accordingly we have the general result that, when tubes of different lengths are divided into the same number of ventral segments, their times of vibration are proportional to the lengths of the tubes, or, which comes to the same thing, their rates of vibration inversely loroportional to their lengths. The reader should observe that it has been through- out this discussion assumed that the Quaterial, thick- ness, and tension of the tube, or tubes, in question, were subject to no variation whatever. Any changes in these would correspondingly affect the rates of vibration produced. Y. § 54.] MOTIOX OF SOUXDIXG STRIXG. 105 5 4. We are now prepared to examine tlie motion of a sounding string. Its ends are fastened to fixed points of attachment and tlie string is excited at some intermediate point, by plucking it with the finger, as in the harp and guitar, by striking it with a soft hammer, as in the pianoforte, or by stroking it with a resined bow, as in the violin and other instru- ments of the same class. The impulses thus set up are reflected at the extremities of the string (in the violin at the bridge and at the finger of the per- former) and behave towards each other exactly as in the case of the vibrating tube considered above. The results thus obtained are therefore directly applicable to the case before us. The string may vibrate in a single segment as in Fig. 26. This is the form of slowest vibration with a string of given length, material and tension. Accordmgly, when thus vibrating, the string produces the deepest note of which, all other conditions remaining the same, it is capable. The string may also vibrate in the forms shown in Figs. 27, 28, 35, or in forms with larger numbers of segments. The rapidity of vibration in any one of these forms is, as we have seen [§ 52], proportional to the number of seg- ments formed, so that, with two segments, it vi- brates twice, with three, thrice, with ^ovoc^four times, as fast as in the form with one segment. It follows lOG TOIXT OF FERCUSSIOX. [Y. § 54. lieiice [§ 43] that the notes obtained by cansing a string to vibrate successively in forms of vibration with 1, 2, 3, 4, 5 &c., segments are all partial-tones of one compound sound, the lowest being of course its fundamental-tone. The modes of elicitincr the sounds of strino;ed instruments described on p. 105 are not capable of setting up any one of the above forms of vibration hy it self J but cause several of them to be executed to- gether. The result is that each form of vibration called into existence sings, as it were, its own note, without heeding what is being done by its fellows. Accordingly, a certain number of tones belonging to one family of partial- tones are simultaneously heard. What precise members of the general series of partial-tones [p. 84] are present, and with what relative intensities, in the sound of a string set vibrating by a blow, depends on the position of the point at which the blow is delivered, on the nature of the striking- object, and on the material of the string. It is clear that a node can never be formed at the point of percussion. Therefore no partial- tone requiring for its production a node in that place can exist in the resulting sound. If, for in- stance, we excite the string exactly at its middle point, the forms of vibration with an even number of ventral segments, all of which have a node at the V. § 54.] MODE OF FERCUSSIOX. 107 centre of tlie string, are excluded, and only the odd partial-tones, i.e. the 1st, 3rd, 5th, and so on, are heard. In this manner we can always prevent the formation of any assigned partial-tone, by choosing a suitable point of percussion. On the other hand, a vibration-form is in the most favourable position for development when the middle point of one of its ventral segments coincides with the point of per- cussion. The more nearly it occupies this position the louder will be the corresponding partial-tone, while the more it recedes from this position towards that in which one of its nodes falls on the point of percussion, the weaker will the partial-tone be- come. The form and material of the hammer, or other object with which the string is struck, have also a great influence in modifying the quality of the sound produced. Sharpness of edge and hardness of sub- stance tend to develope high and powerful over- tones, a rounded form and soft elastic substance to strengthen the fundamental-tone. The material of the string itself produces its effect chiefly by limiting the number of partial-tones. The stiffness of the string resists division into very short seg- ments, and this implies, for every string, a fixed limit beyond which further submission becomes im- possible ; and where, therefore, the series of over- 108 PITCH OF STRIXG-SO VXDS. [V. § 5-5. tones is cut short. Hence very tliin mobile strings are favourable, thick weighty strings unfavourable, to the production of a large number of partial-tones. . 55. Having examined what determines the qualitij of the sound of a vibrating string, we have next to enquke on what its 'pitch depends. This term is indeed, strictly speaking, inappropriate to a compo- site sound containing a series of different tones, each having its own vibration-number and definite posi- tion in the musical scale. If, however, we use the phrase 'pitch of a sound' as equivalent to 'pitch of the fundamental tone of the sound,' we shall avoid any confusion arising from this circumstance. The pitch of a string-sound depends of course on the rate at which the string is vibrating. We have seen that, when the material thickness and tension of a string remain the same, its jate of vibration varies inversely as the length of the string. Accordingly, the vibration-numhcr of a string-sound varies inversely as the length of the string. It follows hence that the numerical relations between the vibration-numbers of sounds forming given intervals with each other, hold equally for the lengths of the strings by which such sounds are produced. To verify this by experi- ment we have only to stretch a wire between two fixed points A and jB, and divide it into two seg- ments by applying the finger to it at some inter- y, §55.] LENaTEB AND INTERVALS. 109 mediate point (7. li AC bears to CB any one of the simple numerical ratios exliibited in the table on H B p. 66, we obtain the corresponding interval there given by alternately exciting the vibrations of the two segments at any pair of points in AC and CB respectively. Thus, if CB is twice as long as A C, the sound produced by the former will be one octave lower than that produced by the latter. If ^4 (7 is to CB in the proportion of 2 to 3, AC's sound will be a Fifth above CB'b; and similarly in other cases. It was by experiments of this kind that the ancient Greek philosopher, Pythagoras, discovered the exist- ence of a connection between certain musical intervals and the ratios of certain small integers. He ascer- tained that an octave was produced by a wire divided into two parts in the proportion of 2 to 1 ; that a Fifth was obtained by division in the proportion of 3 to 2, and so forth. The relations existing between these leno^ths and the vibration-numbers of the notes produced by them were entirely unknown to Pytha- goras and his contemporaries ; indeed it was not until the seventeenth century that they were dis- covered by Galileo. 110 THE PIANOFORTE, [Y. § :jC>. In instruments of tlie violin class, the pitch of the notes sounded varies with the position of the finger on the vibrating string. The length of string intercepted between the fixed bridge and the finger admits of being altered at pleasure, and thus every shade of pitch can be produced from such instruments. The resined bow maintains the vibration of the string by alternately dragging it out of its position of rest, letting it fly back again, catching it once more, and so on. The hollow cavity of the instrument rein- forces the string-sound by resonance. The quality of instruments of the violin class is vivacious and piercmg. The first eight partial tones are well represented in their clang. The Pianoforte. « 56. In this instrument each wire is stretched between two pegs, which are fixed into a flat plate of wood called the sound-board. The string is fas- tened to one peg, and coiled round the other, which admits of being turned about its own axis by means of a key of suitable construction. In this manner the strmg can be accurately tuned, since by tighten- ing or loosening the wire, we raise or lower its pitch at pleasure. In small instruments two, in larger ones three, wires in unison with each other usually correspond to each note of the key-board. While y. §56.] ACTION- OF PEDALS. Ill tlie instrument is not in action a. series of small pieces of wood covered with list, called 'dampers,' rest upon the wires. These are connected with the key-board in such a manner that, when a note is pressed down, the corresponding damper rises from its place, and the wires it previously covered remain free, until the note is allowed to spring up again, when the damper immediately sinks back into its original position. Each note is connected with an elastic hammer, which deals a blow to its own set of wires, and then springs back from them. The wires thus set in motion continue to vibrate until either the sound gradually dies away, or is abruptly extm- guished by the descent of the damper. The action of the two pedals is as follows : the soft pedal shifts the key-board and associated hammers m such a way that each hammer only acts on one of the wires coiTesponding to it, instead of on its complete set of two or three wires. The sound produced by striking a note is therefore proportionally weakened. The loud pedal lifts all the dampers off the wires at once. It thus not only allows notes to continue sounding after the finger of the player has quitted them, but places other wires than those actually struck ui a position to sound by resonance. The number of wires thus brought into play by striking a smgle note of the instrument will be easily seen to be con- 112 EFFECT OF LOUD PEDAL. [V. § 56. siderable. Suppose, first, that a simple-tone, e.g. that of a tuning-fork, is sounding near the wires of a pianoforte with the loud pedal doivn, its pitch being that of middle C, ^ ; the wires of the corre- sponding note will of course resonate with it, vibrat- ing in the simplest form with only one ventral seg- ment. The wires of the note ^zE^z one octave below it, are also cajDable of producing middle C when they vibrate in the form with two segments. So are those of ^ , a Twelfth below it, when ~~ OE vibrating with three segments, those of — :, two octaves below it, -vibrating with four segments, and so on. Proceeding in this way we determine a series of notes on the key-board of the pianoforte, the wires of which are able to produce a simple tone of the pitch of middle C. They obviously follow the same law as the harmonic overtones of a compound sound with middle C for its fundamental-tone, except that the successive intervals are reckoned cloivnwards instead of upwards. The wires of all these notes will reinforce the tone of the tuning-fork by reso- nance. If now we remove the fork, and strike middle C on the pianoforte itself, we obtain, of course, a compound sound consisting of a number of simple tones. To each of these latter there corresponds a Y. § 56.] QUALITY OF PIANOFORTE SOUNDS. 113 descending series of notes on the key-board, com- mencing with that whose fundamental is in unison with the simple-tone in question. A full chord struck in the middle region of the instrument will, in this way, command the more or less active services of two or three times as many wires as have been set vibrating by direct percussion. The increase of loud- ness thus secured is not very considerable, the effect being rather a heightened richness, like that of a mass of voices singmg pianissimo. The actual inten- sity of the sound so heard may be less than could be produced by a quartett of solo singers, but it possesses a multitudinous character which the other lacks. The sustaining power of the loud pedal renders care in its employment essential. It should, as a general rule, be held down only so long as notes belonging to one and the same chord are struck. Whenever a change of harmony occurs, the pedal should be allowed to rise, in order that the descent of the dampers may at once extinguish the preceding chord. If this precaution is neglected, perfectly irreconcilable chords become promiscuously jumbled together, and a series of jarring discords ensue, which are nearly as distressing to the ear as the striking of actual wrong notes. The quahty of pianoforte notes varies greatly in different parts of the scale. In the lower and middle region it is full and rich, the first six T. 8 lU ACTION OF SOUND-BOARD. [V. § 50. partial-tones being audibly present, though 4, 5, 6 are much weaker than 1, 2, 3. Towards the upper part of the instrument the higher partial-tones dis- appear, until in the uppermost octave the notes are actually simple-tones, which accounts for their tame and uninteresting character. The pianoforte shares with all instruments of fixed sounds certain serious defects, which will be discussed in detail in a subse- quent chapter. When a vibrating wire is passing through its undisturbed position, its tension is necessarily some- what less than at any other moment, since, in order to assume the curved segmental form, it must be a little elongated, which involves a corresponding in- crease of tension. Hence the two pegs by which the ends of a wire are attached to the sound-board are submitted to an additional strain twice during each complete segmental vibration. The sound-board, being purposely constructed of the most elastic wood, yields to the rhythmic impulses acting upon it, and is thrown into segmental vibrations like those of the wire. These vibrations are communicated to the air in contact with the sound-board, and then transmitted further in the ordinary way. The amount of surface which a wire presents to the air is so small, that, but for the aid of the sound-board, its vibrations woidd V. § 57.] SOUJ^DS OF ORGAN PIPES. 115 hardly excite an audible sound. The reader will not fail to notice that the sound-board of the pianoforte plays the same part as the hollow cavity of the violin, and is, in fact, a solid resonator. In the harp, the framework of the instrument serves the same purpose. We ha.ve, ui this combination of a vibration-exciting apparatus with a resonator, the type of construction adopted in nearly all musical instruments. 3. Sounds of organ-pipes^ 57. It has been shown [§ 51] that, when two series of equal waves due to transverse vibrations, travel along a stretched wire, in opposite directions, stationary nodes are formed at equal distances along it, separated by vibrating segments of equal lengths. Let us now suppose that two series of equal waves due to longitudinal vibrations are traversing, in oppo- site directions, a column of air contained in a tube of uniform bore. Each set of such waves has its own associated wave-form [§ 18]. These will bsLave to each other precisely in the same way as the trans- verse waves of Fig. 35. We have only, therefore, to consider the curves drawn in that figure as con- stituting the associated waves for the longitudinal air-vibrations, in order to make the conclusions of § 52 at once applicable to the case before us. The 8—2 116 'STOPPED' AXD 'OPEN' OPcGAy PIPES. [T. § 58. result is a series of equidistant nodes, or points of permanent rest, distributed along the column of air. The intervening portions of air vibrate longitudinally at the same rate as the corresponding ventral seofments of Fi^. 35. We have here, as in the case of the sounding wire, all the conditions for the production of a musical note, of pitch correspond- ing to the rapidity of vibration obtained. It only remams to show that, in the case of every organ- pipe, two sets of equal waves traverse in opposite dhections the air-column which it contains. Organ pipes are of two kinds, called respectively ' stopped ' and ' open,' — epithets which, however, apply only to one end of the pipe ; the other is in both kinds open. To begin with the first variety. 58. Let AB, Fig. 37, be the closed end of a stopped pipe, and let a series of pulses of condensa- tion and rarefaction be passuig along the air -vvithin it, hi the dhection sIiot^tl by the arrow. First let a pulse of condensation, CABD, have just reached AB. By supposition, the air in AD is denser, and there- T. § 58.] REFLECTION AT A CLOSED ORIFICE. 117 fore at a higlier pressure, tlian tlie air behind it. It will therefore expand. Forward motion being barred by AB, the expansion must take place entirely in the opposite direction. Hence the pulse of conden- sation is reflected at the end of the pipe, and pro- ceeds to traverse its previous course in the reverse direction. Next, suppose CABD to be a pulse of rarefaction. The air in it is at a less pressure than that of the air behind it. Accordingly, it will be condensed between the pressure from behind and the resistance of the fixed obstacle in front. The con- densed pulse behind it will expand during the pro- cess and become itself rarefied. Thus a pulse of rarefaction, equally with one of condensation, is re- flected at the closed end of the pipe. Neither pulse suSers any other change except of direction of mo- tion. Since every pulse is thus regularly reflected at AB, and made to travel back unchanged along the pipe, it follows that a system of equal waves advancing in the direction of the arrow is necessarily met by an exactly equal system proceeding in the opposite direction. For stopped pipes, therefore, the point required to be proved is made out. Let AB, Fig. 38, be one end of an open pipe, along which condensed and rarefied pulses are being alternately transmitted in the direction of the arrow. First, let CABD be a pulse of condensation which 118 REFLECTION AT AN OPEN ORIFICE. [Y. § 58. has just reached AB. The air in it is at a higher pressure than the outer air beyond ^^^ which is in the ordinary atmospheric condition, neither condensed nor rarefied. Hence some of the advanced part of the pulse CABD will escape into the open air. This exit will cease as soon as the air just in front of AB has been sufficiently condensed by its means. But, in the mean time, CABD has become rarefied by the escape of part of its air. Hence a rarefaction will travel back along the tube. Now, let CABD have been originally a rarefied pulse. It will be converted by the superior pressure of the air, both in front and rear, into a condensation, and in this condition start on its backward route. By the above reasoning^, which the student should carefully compare with that of [§ 21], it is clear that reflection takes place at an open, as well as at a closed end of a pipe ; with this difference however, that in the former case condensation is turned into rarefaction and rarefaction into conden- ^ I am indebted for this popular explanation of reflection at the open end of a pipe to Mr Coutts Trotter, Fellow and Tutor of Trinity College, Cambridge. V. §59.] YIBr.ATI0X-F0B2IS FOR STOPPED PIPE. 119 sation, so tliat tlie wave returns hind part before. We liave tliiis established for an open pipe what was proved for a stopped one on p. 117» 59. We will now examine what forms of seof- mental vibration the air in a stopped pipe can adopt. Every sucli form must necessarily have a node coin- cident icith the closed end of the pipe, since no longi- tudinal vibrations are possible there. The impulses constituting the series of direct waves are not, as we shall see presently,, originated, like those of a piano- forte string, at some intermediate point, but enter the pipe at its open end. This must therefore be a point of maximum vibration. Now a glance at Fig. 35, shows that the maxima of vibration are at the middle points of the ventral segments. Hence the centre of a segment must coincide ivith the open end of the pipe. The above considerations suffice to solve the problem before us. If the closed end of the pipe is placed at A (Fig. 35), the open end must be midway between A and 1, or between 1 and 2, 2 and 3, 3 and 4, and so on. No other forms of vibration are possible. Fig. 39 shows the air in a stopped pipe of given length vibrating in four such ways. The vertical Hues indicate the positions of the nodes. For the sake of greater clearness, the loops of the 120 VIBRATION-FORMS FOR STOPPED PIPE. [Y. § 59. associated vibration-forms are in eacli case drawn in dotted lines. (a-) 2%, 39, (b) (0 (°) In (A) we have half a segment; in {B) a segment and a half; in (C) two segments and a half; in (D) three segments and a half The numbers of seg- ments into which the length of the air-column is divided, in the four cases, are, therefore, proportional to 1 11, 21 and 31, i.e. to J, f, |, and f, or to the whole numbers 1, 3, 5, and 7. Now by § 53 it appears that the rate of vibration in any form varies as the number of segments into which it is divided. The vibration-numbers of the sounds produced in the present instance are, therefore, Y. § 60.] VIBRATION'-FORMS FOR OPEN PIPE. 121 proportional to 1, 3, 5 and 7, i. e. we get the first four odd partial-tones of a sound of which [A) gives us the fundamental tone [§ 43]. The reasoning here adopted evidently applies equally well to cases in which the air- column is subdivided to any assigned extent. It follows, therefore, that the notes obtain- able from a stopped pipe are all odd partial-tones belonging to one and the same clang. 60. The case of the open pipe shall next be investigated. Here, as in the previous case, the end at which the direct pulses enter must be at the centre of a ventral segment. The considerations alleged on p. 118 indicate that the same thing must also hold good at the opposite orifice. Referring once more to Fig. 35, we obtain all the possible modes of vibration which satisfy both the above conditions by placing one end of the pipe mid- way between^ and 1, and the other successively half way between 1 and 2, 2 and 3, 3 and 4, and so on. The first four of the cases thus obtained, for a tube of constant length, are shown in the next figure, which is drawn on precisely the same plan as Fig. 39. In each case, the two half segments at the ends of the pipe make up one whole segment. The num- bers of segments into which the air- column is divided are, therefore, in {A), 1 ; in {B), 2 ; in (C), 3 ; in [D), 4. The same law would obviously hold for 122 VIBBATI0X-F0R2IS FOR OPEN PIPE. [V. § Gl. higlier subdivisions. Hence, in the case of an open pipe, the rates of all the possible modes of segmental Fig. 40. (^) (p) (O (o) ... ■:- ,.-- 2> < "^> ~^"4C 2> ,.-- -H--' H'-"' ■•-...^ .,....----., --'1 ----.. vibration are as the numbers 1, 2, 3, 4, 5, &c. The notes obtainable from such a pipe are, therefore, the complete series of partial-tones belonging to one and the same clang. 61. If the sloivest forms of vibration, shown at [A) in Figs. 39 and 40, are comj)ared with each other, it will be at once seen that the vibrating segment of Fig. 39 is exactly twice as long as that of Fig. 40. Hence, the deepest tone ohtainahle from a stopped pipe is ahvays precisely one octave loiver than the gravest tone proditcihle from an open p)ipe of the same length. It has been shown in § 39 that this result of theory is borne out by experiment. Y. § 62.] PITCH AND LENGTH OF PIPE. 125 62. In order to complete this investigation, it is necessary to determine the pitch of the lowest note which a pipe of given length is capable of uttering. By § 52 we know that a complete segmental vibra- tion is performed during the time occupied by a pulse in traversing tivice the length of a single seg- ment. In (A) Fig. 39, this is equal to four times the length of the tube. The velocity of the pulse is here the velocity of Sound in air, which, under ordinary conditions of temperature, &c., we may put at 1125 feet per second-^. The vibration-number of a stopped pipe's lowest tone is therefore found by dividing 1125 by four times the length of the pipe expressed in feet. Conversely the length of a stopped pipe, which is to have as its deepest tone a note of given pitch, is found by dividing 1125 by four times the vibration-number of the note to be produced. The quotient gives the required length in feet. For example, middle C of the pianoforte makes 264 vibrations per second. The required length in this 1125 case would be expressed by t— r-r, which is rather lOob more than 1 ft. |in., i.e. roughly speaking, one foot. An open pipe, to produce the same note, would there- fore have to be two feet in length. It has been just shown that the vibration-number ' Tyndall's Sound, p. 24. 124 PITCH AND LEXGTU OF PIPE. [Y. § 63. of the lowest tone producible, either from a stopped or an open pipe, varies inversely as the length of the pipe. The length of the pipe therefore varies in- versely as the vibration-number. Hence the rela- tions established in § 55 for strings, hold also for columns of air contained in pipes. The case of the pipe-sounds is, however, somewhat simpler than that of the string-sounds, since the pitch of the latter depends on the tension of the strings as well as on their lengths, whereas, in the former, pitch depends, under given atmospheric conditions, on length alone. Hence we may define a note of assigned pitch by merely stating the length of the stopped or open pipe, w^hose fundamental tone it is. The open-pipe is commonly preferred for this purpose, and accord- ingly organ builders call middle C ' 2 foot tone ; ' the octane below it '4 foot tone,' and so on. The lowest C on modern pianofortes is ' 1 6 foot tone ; ' that one octave lower, which is found only on the very largest organs, '32 foot tone.' The highest note of the pianoforte, usually A, would be about '2 inch tone/ 63. The reader should observe that, in the course of this discussion, we have incidentally obtained a more complete theory of resonance tlian could be given in chapter iii. When a tuning-fork is held at the orifice of a tube, the strongest resonance will be produced if the note of the fork coincides with the V. §§ 64, 65.] CLASSIFICATION OF ORGAN PIPES. 125 fundamental tone of the tube. A decided, thouofh less powerful, resonance ought also to ensue if the fork-note coincides with one of the higher tones of the tube, which, as we know, are all overtones of its fundamental. A resonance-box is only a stopped pipe under another name. We may therefore employ it to test the truth of our result, that the only tones obtainable from a stopped pipe are the odd partial- tones of a clang, of which the first is the fundamental tone. I possess a series of forks giving the first seven partial-tones of a clang. When I strike 1, 3, 5 or 7, and hold them before the open end of the resonance-box corresponding to 1, a decided rein- forcement of their tones is heard. If I do the same with 2, 4, or 6, hardly any resonance is produced. Thus our theoretical result is experimentally verified. 64. Organ pipes are divided into two classes according as the soiuids, which they are to strengthen by resonance, are originated (1) by blowing against a sharp edge, (2) by blowing against an elastic tongue. Those of the first-class are called ^?^e-pipes ; those of the second class i'eec^- pipes. We will consider each class by itself 65. Flue-pipes. Here the wind is driven through a narrow slit against a sharp edge placed exactly opposite to it, in the manner shown in Fig. 41, which 126 CONSTRUCTION OF FLUE-PIPES. [V. § ^5. represents a vertical section of a portion of the pipe near the end at which its sound originates. i^.41, The air is forced by the bellows tlirough the tube ah, into the chamber c, and escapes through the slit cZ, thus impinging against the edge e, where it pro- duces a sharp hissing sound wliich may be imitated by blowing with the mouth against a knife-edge held in front of it. This sound, as we shall see in the sequel, may be regarded as consisting of a great variety of notes of different pitch. Of these the pipe is able to reinforce, by the resonance of its air-column, such notes as coincide with its own essential tones. The quality of the sound thus resulting will, of course, depend on the number, orders and relative intensities of the partial-tones present in the clang heard. The origmal hissmg sound contributes nothing directly to the whole V. §65.] QUALITY OF SOUNDS OF FLUE-PIPES. 127 effect, being, with well constructed meclianism, in- audible except close to the pipe. .; Stopped wooden Hue-pipes of large aperture, blown by only* a light pressure of wind, produce sounds which are nearly simple tones; only a trace of partial-tone No. 3 being perceptible. Such tones, like the fork tones with which they are in fact almost identical, sound sweet and mild, but also tame and spiritless. A greater pressure of wind developes 3 distinctly, in addition to 1, and, if it becomes excessive, may spoil the quality by giving the overtone too great an intensity compared to that of the fundamental, or may even extinguish the latter altogether, and so cause the whole sound to jump up an octave and a Fifth. This result may easily be obtained by blowing with the mouth into a small 6-inch stopped pipe, which can easily be ob- tained at any organ factory. Stopped pipes of narrow aperture develope 5 audibly, as well as 1 and 3. In the case of an open pipe the fundamental-tone is never produced by itself According to the dimen- sions of the pipe, and the pressure of wind, it is accompanied by from two to ^yq overtones. Open flue-pipes present, therefore, various degrees of timbre which are exhibited in the different 'stops' of a large organ. 128 MECHANISM OF A 'BEED: [Y.^QQ. Q,Q. Reed-pipes. The apparatus by wliicli the sounds of pipes of this class are originated is the following. One end of a thin narrow strip of elastic metal, called a 'tongue/ is fastened to a brass plate, while the other end is free. A rect- angular aperture, very slightly larger than the tongue, is cut through the plate, so as to allow the tongue to oscillate into and out of the aperture, Hke a door with double hinges, without touchino* the edges of the aperture as it passes them. The accom- panymg figure shows this piece of mechanism, which is called a 'reed,' in its position of rest. Fig.42. It is set in motion by a current of air being driven against the free end of the tongue, which is thus made to swing between limiting positions as shown in the annexed sections. ^^^ (A) ^^-^3. (^^ iiiii ^^~ pii ilill'^ — iiil When the tongue occupies a position intermediate between that of {A) and its position of equilibrium, the air passes through the aperture in the direction indicated by the arrows in (A). At the moment that the tongue passes through its equilibrium position V. §6G.] QUALITY- OF REED^SOUXDS. 120 towards that sliown in [B], the current of air is barred bj the accuracy with which the tongue fits into the aperture beneath it. Only when the tongue again emerges can the air resume its passage. The reed thus produces a series of equal discontinuous impulses of air at equal intervals of time. The principle of the instrument is identical with that of the Syren, and it therefore gives rise to a regular musical sound. Its note is a highly composite clangs containing dis- tinctly recognizable partial-tones up to the 16th or 20th of the series. Thus a reed does not require to be associated with a resonating column in order to produce a musical sound ; in fact the instrument called the harmonium consists of reeds without such adjuncts. The timbre of an independent reed is, however, characterised by too great intensity on the part of the higher partial-tones. It is desirable to correct this defect by strengthening the fundamental- tone of the clang. This is done by placing the reed in the mouth of a pipe Avhose deepest tone coincides with the fundamental-tone of the reed-clang. This tone will then be most powerfully reinforced by resonance. The other partial-tones of the clang (the odd ones only in the case of a stopped pipe) will also be strengthened by resonance, but to a smaller and smaller extent as their order rises. The force required to throw a column of air into rapid vibra- T. 9 130 SOUNDS OF REED-PIPES [Y. § C6. tion is greater than suffices to set up a slow vibra- tion. Hence, if two partial-tones in the reed-clang were exactly equally intense, the lower of them would cause a more powerful resonance than the higher. Since the force necessary to produce seg- mental vibration increases very rapidly, as the sub- divisions of the air-column become more numerous, the very high partial-tones of the reed-clang are practi- cally unsupported by the resonance of the associated pipe. It will be seen hereafter how the quahty of the resulting sound is improved by this circumstance. It is clear that sounds differing widely in quality may be obtained by associating a reed with pipes of different lengths and forms. If the pipe's funda- mental tone coincides with that of the reed-clang, in the case of a stopped pipe, only odd, in that of an open pipe, both odd and even, partial-tones are strengthened by resonance. If the fundamental tone of the pipe coincides with one of the overtones of the reed-clang, the quality of the resulting sound is correspondingly affected. The form of the pij^e may also be modified, so as to be conical, or of any other shape, which will bring in other changes in its resonating properties. In these ways we have pro- vision for the great variety of quality among reed- pipes, which we find represented in organ stops of that class. V. §§ 67, 68.] ORCHESTRAL WIXD-IXSTRUMENTS. 131 4. Sounds of orchestral wind-instruments and of the human voice. G7. The fiute is in principle identical with an open flue-pipe. The lips, and a hole near the end of the tube, play the parts of the narrow slit and opposing edge. The qualitj of the instrument is sweet, but too nearly simple to be heard during a long solo without becoming wearisome. Its most lovely effects are produced by contrast with the more brilliant timhre of its orchestral colleaofues. The clarionet, hauthois and bassoon have wooden reeds. The clarionet has a stopped cylindrical tube, producmg only odd partial-tones, whence its cha- racteristic quality. The hautbois and bassoon have conical tubes. In the horn and trumpet the lips of the j^er- formers supply the place of a reed. 68. The apparatus of the human voice is essen- tially a reed (the vocal chords), associated mth a resonance-cavity (the hollow of the mouth). The vocal chords are elastic bands, situated at the top of the wind-pipe, and separated by a narrow slit, which opens and closes again with great exact- ness, as air is forced through it from the lungs. The form and width of the slit allow of being quickly and extensively modified by the changing tension of 9—2 m MECHANIS:^! OF THE EVMAK VOICE. [Y. § GS. the vocal chords, and thus sounds widely differing from each other in pitch may be successiyely pro- duced with surprising rapidity. In this respect, the human * reed ' far exceeds any that we can arti- ficially construct. The size and shape of the cavity of the mouth may be altered by openmg or closing the jaws, raising or dropping the tongue, and tightening or loosening the lips. We should expect that these movements would not be without effect on the reso- nance of the contained air, and such proves on experiment to be the fact. If we hold a vibrating tuning-fork close to the lips, and then modify, suc- cessively, the resonating cavity, in the ways above described, w^e shall find that it resounds most power- fully to the fork selected when the parts of the mouth are in one definite position. If we try a fork of different pitch, the attitude of the mouth, for the stroncxest resonance, is no lonofer the same. Hence, w4ien the vocal chords have originated a reed-clang containmg numerous well developed partial-tones, the mouth- cavity, by successively throwing itself into different postures, can favour by its resonance, first one partial-tone, then another ; at one moment this group of partial-tones, at another tliat. In this manner endless varieties of qualitv are rendered possible. The art of vocalizing consists V. §G9.] OVERTOXES OF THE IIUMAX VOICE. 133 ill so placing the resonating apparatus of tlie voice as to modify the clang due to the vocal chords in the way most attractive to the ear. The complete analysis of the sounds of the hu- man voice into their separate partial-tones presents peculiar difficulties to the unassisted ear, and can hardly be effected without the help of resonators such as those described in § 42. By their aid we can detect in the lower notes of a bass voice, when vigorously sung, shrill overtones reaching as far as No. 16, which is four octaves above its fundamental- tone. Under certain conditions these high overtones can be readily heard without recourse to resonators. When a body of voices are ringing fortissimo without any instrumental accompaniment, a peculiar shrill tremulous sound is heard which is obviously far above the pitch of any note actually being sung. This sound is, to my ear, so intensely shrill and piercing as to be often quite painful. I have also observed it when listenino^ to the lower notes of an unusually fine contralto voice. The reason why these acute sounds are tremulous will be given later. 69. We close this discussion by describing a mode of submitting Helmholtz's general theory of musical quality to a further, and very severe test. The sounds of tuning-forks when mounted on their appropriate resonance-boxes are, as we know, 134 COALESCENCE OF PARTIAL-TONES. [Y. % G9. very approximately simple tones. If, therefore, we allow a number of such sounds, coincident in pitch with the fundamental-tone, and with individual overtones, of one and the same clang, to be simul- taneously produced, the effect on the ear ought, if Helmholtz's theory is true, to be that of a single musical sound, not that of a series of independent tones. To try the experiment in the simplest form, take two mounted forks forming the interval of an octave, and cause them to utter their respective tones together. For a short time Ave are able to dis- tinguish the two notes as coming from separate instruments, but soon they blend into one sound, to which we assign the pitch of the loiver fork, and a quality more brilliant than that of either. So strong is the illusion, that we can hardly believe the higher fork to be really still contributing its note, until we ascertain that placing a finger on its prongs at once changes the timbre , by reducing it to the dull, uninteresting quality of a simple tone. The character of a clang consisting of only one overtone and the fundamental, may be shown to admit of many dif- ferent shades of quality, by suitably varying the relative intensities of the two fork-tones in this experiment. If we add a fork a Fifth above the higher of the first two, and therefore yielding the third partial-tone of the clang of which they form Y. § 69.] SYNTHETIC APPARATUS. 135 the first and second, the three tones blend as per- fectly as the two did before ; the only difierence perceptible being an additional increase of brilliancy. The experiment admits of being carried further with the same result. If we were able to produce by means of tuning- forks as many simple tones of the series on p. 84 as we pleased, and ako to control at will their relative intensities, it would be possible to imitate, in this manner, the varying timbre of every musical instru- ment. The unmanageable character of very high forks has as yet prevented this being done for sounds containing a very large number of powerful over- tones, but an apparatus on this principle has been devised by Professor Helmholtz, which imitates, very successfully, sounds not involving more than the first six or eight partial-tones. His theory of quality is thus experimentally demonstrated, both analyti- cally and synthetically. We will examine in the next chapter some important theoretical considera- tions by which this theory is further elucidated and confirmed. CHAPTER YI. ON THE CONNECTION BETWEEN QUALITY AND MODE OF VIBEATION. 70. It was stated on p. 71 that, wlien a pendu- lum performs oscillations whose extent is small com- pared to the length of the pendulum itself, the period of a vibration is the same for any extent of swing ivithin this limit. We will apply this fact to prove that the prongs of a tuning-fork vibrate in the same mode (§ 11) as does a pendulum. When a sustained simple tone is being trans- mitted by the air, we may regard it as originated by a tuning-fork of appropriate pitch and size. But we know experimentally that, by suitable bowing, we may elicit from such a fork tones of various de- grees of intensity, though having all the same pitch. Here, therefore, the extent of vibration varies, while the 2^eriod remains constant, which is the pendulum- law. Accordingly the vibrations of a tuning-fork are identical, in mode, with those of a pendulum. The same thing will hold good of the aerial vibra- tions to which those of a fork give rise. Hence, in general, a simple tone is due to vibrations executed according to the pendulum-law. YL%n.]s(;pi:FcFOSiTioy of small motions. 137 Such vibrations when performed longitudinally, will, therefore, give rise to waves of condensation and rarefaction whose associated wave-form is that drawn in Fig. 17 {a) p. 38. It will be convenient to call the vibrations.to which a simple tone is due simple vibrations; and the associated waves simple luaves. We proceed to examine the modes of vibration cor- responding to composite sounds. Let us, first, take the case of a sustained clang consisting of but two simple tones, the fundamental and its first overtone. A particle of air engaged in transmitting this sound is simultaneously acted upon by two sets of vibratory movements, and we have to investigate what its motion will be under their joint influence. In fact, the problem before us is the com- position of two simple vibrations. In order to solve it, we must employ a principle of Mechanics, called the " superposition of small motions," the nature of which can be illustrated experimentally as follows. 71. Suppose a cork to be floating on the undis- turbed surface of a sheet of water into which two stones are thrown at different points. From each origin of agitation concentric circular waves will spread out, and presently the cork will be influenced by both sets of disturbances at once. Either series of waves, if it acted separately on the cork, would cause it to execute a vibratory movement in a verti- 138 SUPERPOSITIOX OF S2fALL MOTIONS. [VI. § 71. cal straight line. The mechanical principle which we are explaining asserts that the joint effect of the two sets on the cork will be exactly equal to the sum or difference of their separate effects, according as these are produced in the same, or in opposite directions. The accompanying figures show the four different cases which may arise. In each, a and h are the j)oints which the cork, originally at rest at O in the level-line, would occupy, at the moment indicated in the figure, were it acted on by either set of waves alone ; c is its contemporaneous position under their joint action. (1) Fi^.44. (2) In (1), crest falls on crest ; in (2), trough on trough, and the displacement, Oc, of the cork from its position of rest, 0, is equal to the sum of the dis- placements due to the two crests separately, viz. Oa and Oh. In (3) and (4), where the crest of one wave meets the trough of another, Oc is equal to YI. §72.] DETERMINATION OF JOINT WAVE. 139 the difference between Oct and Oh ; c being above or below tlie level-line, according as Oa is greater than Oh, as in (3), or less than Oh, as in (4). Thus, each wave produces its own full effect on the cork in its own direction, or, in other words, the motion due to one wave is ' superposed ' on that due to the other. In order, then, to determine the form of the joint wave Avhich results from the combination of two con- stituent waves, we have only to apply the above principle successively to points in the level-line which both sets of disturbances simultaneously affect. We thus obtain an assemblage of points constituting the joint wave required. In the instance now before us we proceed as follows. Let each simple tone be represented by its associated wave-system. Ascertain by the process just described to what joint form the combination of the two associated wave-systems lead. The result will be the associated system corresponding to the mode of particle-vibration to which the compound sound is due. 72. Before, however, we can lay down the two tributary systems of waves, an important point re- mains to be settled. We will, for a moment, suppose that the two simple tones on which we are engaged are originated and sustained by two tuning-forks. 140 DIFFERENCE OF PHASE. [VL § 72. situated as in ttie annexed figure, and that we are examining the transmission of their resulting clang along the dotted line with respect to which they are symmetrically situated. Let the forks have been set sounding by pre- cisely simultaneous blows. They will then commence swinging out of their positions of rest i}i the same direction at t\e same instant. The points in the associated wave-forms where a vibrating particle is momentarily in its position of rest, are those in which it cuts the level-line. Hence, in laying dovrn the two tributary wave-systems along the same level-line, we must make them both cut that line in some one point, taking care that their convexities at that point are both turned the same way, as at 0, Fig. 46. In this case the two vibrations are said to start in the same phase. If the two forks are set in vibration at different moments, they may not swing out of their equilibrium VI. QUALITY AXD PHASE. 141 positions in the same direction tog'etlier. Hence we no longer necessarily have a point where both sets of waves cut the level-line. The result is of the kind shown in Fig. 47, where three different cases are represented. Here we have vibrations starting in differerit phases. It is clear from the figure that all phase- differences can be properly represented by merely causing the wave-systems engaged to assume different positions upon the levehline, tvith reference to each other. The second and third cases are obtainable from the first by sliding the system of shorter waves bodily along the level-line, while the other system of waves retains its position. By the help of the instrument mentioned on p. 135 Professor Helmholtz has demonstrated that, U2 SIMPLE A^D RESULTANT WAVE-FORMS. [VI. § 73. when a number of partial-tones are independently produced, the clang into which they coalesce has the same quality, ivhatever differences of johase may exist among the systems of simple vibrations to ivhich the constituent jpartial-tones are due. Accordingly, we may expect to find that not one single ivaveform, but many such forms, correspond to a sound of given quality and pitch. In Figs. 48, 49, 50, the associated wave-form cor- responding to our clang of two partial-tones (p. 137) is constructed for three degrees of phase-difference. The simple constituent waves are shown in thin, the result of their composition in full lines. In each case two complete wave-lengths of the latter are exhibited. Figs. 51 and 52 present two wave-forms drawn, in the same way, for a clang of constant pitch and quality containing the j)artial-tones 1, 2 and 3. The dissimilarity of form, and therefore of cor- responding particle-vibration, is, in both sets of fio'ures, most marked. 73. It has been shown that, by mere alteration of phase, a very great variety of resultant wave-forms can be obtained from two sets of simple waves of given lengths and amplitudes. Each one of these forms will give rise to a cycle of others, if we allow the relative amplitudes of the constituent systems to be changed, YI. § 73.] SniPLE AXD RESULTANT WA VE-FOEMS. U3 lU VARIETY OF RESULT AXT WAVE-FORMS. [YI4 73. while keeping tlie difference of phase constant. If, therefore, we have at our disposal the systems of F=j.5l. JEYg . 52 . simple waves corresponding to an iinhmited number of partial-tones, and can assign to them any degrees of intensity and phase-difference that we choose, it is ma-nifest that we may produce by their combination an endless series of different resulting wave-forms. On the other hand, it is not evident that, even out of this rich abundance of materials, we could build up every form of luave ivhicJi could possibly he assigned. The ^French mathematician Fourier has, however, demonstrated that there is no form of wave which (unless itself sunple) cannot be compounded out of a number of simple waves, whose lengths are in- VI. § 73.] FOURIER S THEOREM. Ud versely as the numbers 1,2, 3, 4, &c. He lias fur- ther shown that each individual wave-form admits of being thus compounded in only one ivay, and has provided the means of calculating, in any given case, how many J and tohat, members of the partial series will appear, their relative amplitudes and their differences of phase. When translated from the language of Mechanics into that of Acoustics, the theorem of Fourier asserts that every regular musical sound is resolvable into a definite number of simple tones whose relative pitch follows the law of the partial-tone series. It thus supplies a theoretical basis for the analysis and synthesis of composite sounds which have been ex- perimentally effected in chapters iv. and v. When we are listening to a sustained clang, the air, at any one point within the orifice of the ear, can have only one definite mode of particle-vi- bration at any one moment. How does the ear behave towards any such given vibration ? It pro- ceeds as follows. If the vibration is simple, it leaves it alone. If composite, it analyzes it into a series of simple vibrations whose rates are once, twice, three times &c. that of the given vibration, in accordance with Fourier's theorem. In the former event, the ear perceives only a simple tone. In the latter, it is able to recognize, by suitably directed and assisted T. 10 1-18 ANALYZING POWER OF THE EAR. [YI. § 73. efforts, partial-tones corresponding to the rate of each constituent into which it has analysed the com- posite vibration originally presented to it. The ear being deaf to differences of ^^/lase in partial-tones (p. 142), perceives no distinction between such modes of vibration as those exhibited on p. 143, but merely resolves them into the same single pair of partial- tones. Since, however, only one such resolution of a given vibration-mode is possible, the ear can never vary in the series of partial-tones to which it re- duces an assigned clang. The power possessed by the ear of thus singling out the constituent tones of a clang, and assigning to them their relative intensities, is unlike any cor- responding capacity of the eye. Take for instance the two curves shown in Figs. 51 and 52, and try to determme, by the eye alone, what simple waves, present with what amplitudes, must be superposed in order to reproduce those forms. The eye will be found absolutely to break down ui the attempt. We have seen that the loudness of a compo- site sound depends on extent of vibration, and its pitch on rate of vibration. There remains only one variable element, viz. mode of vibration, to account for the qualitij of the sound. From this conside- ration it follows that some connection must exist between the quality of a sound and the mode of VI. § 73.] QUALITY AND MODE OF VIBRATION. 147 aerial vibration to wliicL. the sound is due. U^) to the time of Helmholtz no advance had been made in clearing up the nature of this connection. It was reserved for him to show that, while no two sounds of different quaUtj can correspond to the same mode of vibration, many different modes of vibration may yet give rise to a sound of only one degree of quality. In other words, mode of vibration determmes quality, but quality does not determine mode of vibration. 10—2 CHAPTER YII. ON THE INTERFERENCE OF SOUND, AND ON 'BEATS'. 74. In § 71 we examined the principle on which the problem of the composition of vibrations is gene- rally solved. We now approach certain very import- ant particular cases of that problem, which it wiU be worth while to solve both independently, and also as instances of the method repeatedly appHed in § 72. Suppose that a particle of air is vibrating be- tween the extreme positions A and JB, under a A B sustained simple tone prodviced by a tuning-fork, or stopped flue-pipe. Now let a second instru- ment of the same kind be caused to emit a tone exactly in unison ivith the first. We will assume that, when the vibrations constituting the second tone fall on the particle, it is just on the point of starting from A towards B, under the influence of those of the first. Two extreme cases are now possi- ble, depending on the movement which the particle VII. § 74.] INTERFERENCE OF SOUND, 149 would have executed, had it been affected by the later-impressed vibration alone. First, suppose that to be from A along the line AB, either through a greater or less distance than AB, back again to A, and so on. Here the separate effects of the two sets of vibrations will be added together, the particle will, therefore, perform vibrations of larger extent than it would under either component separately. Next, suppose that, under the second set of vibrations alone, the particle would move from A in the opposite direction to its former course, i.e. along BA pro- duced, shown by a dotted line in the figure. In this case the separate effects are absolutely antagon- istic ; accordingly the joint result is that due to the difference of its components. The particle will, therefore, execute less extejisive vibrations than it would have done under the more powerful of the two components acting alone. The most striking result presents itself when the two systems of vibrations, besides being in complete opposition to each other, are also exactly equal in extent. In this case, the air-particle, being solicited with equal intensity in two diametrically opposite directions, remains at rest, the two systems of vibra- tions completely neutralizing each other's effect. In general, however, these systems, even when equal in extent of vibration, are neither in complete opposi- 150 INTERFERENCE OF SOUND, [YII. § 74. tioii nor in complete accordance, but in an interme- diate attitude, so as only partially to counteract, or support, each other. These conclusions admit of being exhibited in a more complete manner by means of associated waves. We have only to lay down the simple wave-forms corresponding to the constituent vibrations, and superpose them as in § 72. The reader will have noticed that the differences of relative motion described on p. 149 are merely phase-differences. (I) In Fig. 54, (1), (2), (3), we have two waves of VII. §75.] INTEBFERENCE OF SOUND. lol unequal amplitudes in comj)lete accordance, complete antagonism, and an intermediate condition respec- tively. In Fig. 55, a case of equal and opposite waves is sliown. In (l) Fig. 54, tlie resultant wave is tlie sum, and in (2) the differ ence of the com23o- nent waves. In (3), we get a wave of intermediate amplitude. These three resulting waves are neces- sarily simple, as otherwise two simple tones in luiison would give rise to a composite sound, which would be absurd. In Fig. 55 the wave-form degenerates into the level-line, i. e. no effect whatever occurs. 7^. Thus, v/hen one simple tone is being heard, we by no means necessarily obtain an increase of loudness by exciting a second simple tone of the same pitch. On the contrary, we may thus weaken the original sound, or even extinguish it entirely. When this occurs we have an instance of a phe- nomenon which goes by the name of Interference. That two sounds slioidd produce cd)solute silence seems, at first sight, as absurd as that two loaves should be equivalent to no bread. This is, however, 152 INTERFERENCE OF SOUND. [YII. § 76. only because we are accustomed to think of Sound as sometliing with an external objective existence ; not as consisting merely in a state of motion of certain air-particles, and therefore liable, on the application of an opposite system of equal forces, to be absolutely annihilated. A single tuning-fork presents an example of this very important phenomenon. Each prong sets up vibrations corresponding to a simple tone, and the two notes so produced are of the same pitch and intensity. If the fork, after being struck, is held between the finger and thumb, and made to re- volve slowly about its own axis, four positions of the fork with reference to the ear will be found where the tone completely goes out. These posi- tions are mid-way between the four in which the faces of the prongs are held flat before the ear. As the fork revolves from one of these positions of loud tone to that at right- angles to it, the sound gradually wanes, is extinguished in passing the Interference-position, reappears very feebly im- mediately afterwards, and then continues to gain strength until its quarter of a revolution has been completed. 76. The case of coexistent unisons has now been adequately examined : we proceed to enquire what happens when two simple tones differing slightly in YII. § 76.] BEATS OF SIMPLE TOXES. 153 pitch, are simultaneously produced. The problem is, in fact, to compound two sets of pendulum-vibrations wliose periods are no longer exactly equal. Let us fix on a moment of time at which the two component vibrations simultaneously soliciting an assigned par- ticle of air are in complete accordance, and suppose that the particle, under their joint influence, is just commencing a vibration from left to right. It will be convenient to call this an outward, and its opposite an imvard swing. Since the periods of the two component vibrations are unequal, one of them will at once begin to gain on the other, and therefore, directly after the start, they will cease to be in com- plete accordance. It is easy to ascertain what their subsequent bearing towards each other wiU be, by considering two ordinary pendulums of unequal periods, both beginning an outward swing at the same instant. Let A be the slower, B the quicker pendulum. When A has just finished its outward swing, B will have already turned back and per- formed a portion of its next inward swing. Thus, during each successive swing of A, B will gain a certain distance upon it. When B has, in this manner, gained one whole swing, i. e. half a complete oscillation, upon A, it will begin an inward swing at the moment ivhen A is commencing cm outivarcl swing. The two vibrations are here, for the moment, in com- 154 BEATS OF SIMPLE TONES. [VII. § 7 G. lolete 02'>position. After another interval of equal length, B, having gained another whole swing, will be one complete oscillation ahead of A, and they will therefore start on the next outward swino^ too^ether, i.e. the vibrations will be momentarily in complete accordance. Thus, during the time requisite to enable B to perform one entire oscillation more than A, there occur the following changes. Complete accordance of vibrations, lasting only for a single swing of the more rapid pendulum, followed by partial accordance, in its turn gradually giving way to discordance, which culminates in complete opposi- tion at the middle of the period, and then, during its latter half, gradually yields to returning accordance, which regains completeness just as the period closes. It follows from this, combined with what is said on p. 149, that in the case of two simple tones, we must hear a sound going through regularly recurring alternations of loudness in equal successive intervals of time, its greatest intensity exceeding, and its least intensity falling short of, that of the louder of the two tones. Each recurrence of the maximum in- tensity is called a heat, and it is clear that exactly one such beat will be heard in each interval of time during which the acuter of the two simple tones performs one more vibration than the graver tone. Accordingly, the number of beats heard in any VII. §77.] NUMBER OF BEATS PER SECOXD. loo assigned time will be equal to tlie number of com- plete vibrations whicli the one tone gains on the other in that time. We may express this result more briefly as follows : the number of heats per second due to tiuo simple tones is equal to the difference of their respective vihration-numhers, 77. By means of the associated wave-forms we can obtain a graphic representation of beats, which will probably be more directly intelligible than any verbal description. In Fig. 56, the constituent simple waves are laid down, and their resultant constructed, for the interval of a semi-tone. The vibration-fraction for this interval is -r ; i. e. l(j 15 vibrations of the higher tone are performed in the same time as 15 of the lower. The figure repre- sents completely the state of things from a maxi- mum of intensity to the adjacent minimum. The time during which this change occurs is one-half of that above-mentioned : accordingly the figure shows 8 and 7h wave-lengths of the respective sys- tems. Thus half a heat is here pictorially repre- sented, the amplitude of the resultant waves steadily diminishing during this period. "We have only to turn the figure upside down, to get a picture of what occurs in the next following equal period. The amplitudes here again increase, until they loG GRAPHIC REPRESENTATION OF BEATS. [YII. §77. regain their former proportions. One whole heat is thus accounted for. YII. § 78.] EXPERIMENTAL STUDY OF BEATS. 157 In addition to the alternations of intensity wliich characterize beats, they also contain variations of pitch. The existence of such variations is both theoretically demonstrable and experimentally re- cognizable, but they are too minute to require ex- amination here^. 78. The most direct way of studying the beats of simple tones experimentally is to take two unison tuning-forks and attach a small pellet of wax to the extremity of a prong of one of them. The fork so operated on becomes slightly heavier than before ; its vibrations are therefore proportionately retarded, and its pitch lowered. When hoth forks are struck and held to the ear beats are heard. These will be most distinct when the fork's tones are exactly equally loud, for in this case the minima of intensity Vill be equal to zero, and the beats will therefore be separated by intervals of absolute, though but mo- mentary, silence. The increase, in rapidity, of the beats, as the interval between the beating-tones widens, may be shown by gradually loading one of the forks more and more heavily with wax pellets, or by a small coin pressed upon them. If it is desired ^ The reader may, if he wishes to pursne this subject, refer to a paper, by the author, on ' Variations of Pitch in Beats , in tlie Philosophical Magazine for July, 1872, from which Fig. 56 has been copied. 158 EXPERIMENTAL STUDY OF BEATS. [VII. § 78. to exliibit these phenomena to a large audience, the forks should first be mounted on their resonance- boxes, and, after the pellets have been attached, stroked with the resined bow, care being taken to produce tones as nearly as possible equal in intensity. Slow beats may also be obtained from any instru- ment capable of producing tones whose vibration- numbers differ by a sufficiently small amount. Thus, if the strmgs corresponding to a single note of the pianoforte are not strictly in unison, such beats are heard on striking the note. If the tuning is perfect, a wax pellet attached to one of the wires will lower its pitch sufficiently to produce the de- sired effect. Beats not too fast to be readily counted arise between adjacent notes in the lower octaves of the harmonium, or, still more conspicuously, in those of large organs. They are also frequently to be heard in the sounds of church bells, or in those emitted by the telegraph wires when vibrating powerfully in a strong Avind. In order to observe them in the last instance, it is best to press one ear against a telegraph-post and close the other : the beats then come out with remarkable distinctness. It should be noticed that, when we are dealing with two composite sounds, several sets of beats may be heard at the same time, if pairs of partial-tones are in relative positions suited to produce them. Thus, VII. § 78.] BEATS OF COMPOSITE SOUiYES 159 suppose tliat two clangs coexist, each of wliicli pos- sesses the first six partial-tones of the series audibly developed. Since the second, third, &c. partial-tones of each clang make twice, three times &c. as many vibrations per second as their respective fundamen- tals (p. 83), it follows that the difference between the vibration-numbers of the two second-tones will be twice, that between those of the two third-tones three times, &c. as great as the difference between the' vibration-numbers of the two fundamentals. Accordingly, if the fundamental tones give rise to beats, we may hear, in addition to the series so accounted for, ^yq other sets of beats, respectively twice, three, four, five, and six times as rapid as they. In order to determine the number of beats per second, for any such set, Ave need only multiply the number of the fundamental beats by the order of the partial-tones concerned. The beats of two simple tones necessarily become more rapid if the higher tone be sharpened, or the lower flattened ; i. e. if the interval they form with each other be widened. The beats may, however, also be accele- rated, without altering the mterval, by merely placing it in a higher part of the scale. In this way greater vibration-numbers are obtained, and the difference of these is proportionally large, though then* ratio to each other remains what it was before. Thus the 160 BEATS OF A GIVEN INTERVAL. [VII. § 78. rapidity of the beats due to an assigned interval depends jointly on two circumstances, the width of the interval, and its position in the musical scale ; in other words, on both the relative and absolute pitch of the tones forming the given interval. CHAPTER VIII. ON CONCORD AND DISCOED. 79. A question of fundamental importance now presents itself, viz. What becomes of beats, ivlien tlieij are so rapid that they can no longer he separately perceived by the ear? In order to answer it, tlie best plan is to take two unison-forks, of medium pitck, mounted on their resonance-boxes, attach a small pellet of wax to a prong of one of them, and then gradually increase the quantity of wax. At first very slow beats are heard, and as long as their number does not exceed four or five in a second, the ear can follow and count them without difficulty. As they become more rapid the difficulty of counting them augments, until at last they cannot be re- cognized as distinct strokes of sound. Even so, however, the ear retains the conviction that the sound it hears is a series of rapid alternations, and not a continuous tone. Its intermittent character is not lost; although the intermittances themselves pass 11 162 CAUSE OF DISCORD. [YIII. § 79. by too rapidly for individual recognition. Exactly the same tiling may be observed in the roll of a side drum, which no one is in danger of mistaking for a continuous sound. Eapid beats produce a decidedly harsh and grating effect on the ear; and this is quite what the analogy of our other senses would lead us to expect. The disagreeable impressions excited in the organs of sight by a flickering unsteady light, and in those of touch by tickling or scratch- ing, are familiar to every one. The effect of rapid beats is, in fact, identical with the sensation to which we commonly attach the name of discord. Let us examine, in somewhat greater detail, the con- ditions necessary for its production between two simple-tones. If we take a pair of middle C unison forks, and gradually throw them more and more out of tune with each other in the way already described, the roughness due to their beats reaches its maximum when the interval between them is about a half- tone : for a whole tone, it is decidedly less marked, and when the interval amounts to a Minor Third, scarcely a trace of it remains. Hence, in order that dissonance may arise between two simple-tones, they must form vv^ith each other a narrower interval than a minor third. If we call this interval the heating- distance for two such tones, we may express the VIII. § 79.] THE ' beating-distance: 163 above condition tlius. Dissonance can arise directly between two simple tones, only ivhen they are within heating -distance of one another. It follows at once from this that the amount of discord beard by no means exclusively depends on the rapidity of the beats produced, since the same interval will give rise to a very different number of beats per second according as it occupies a higb or a low position in the scale. Nevertheless this circumstance exerts a considerable influence in modifying the intensity of the dissonance of given intervals, according to the absolute pitch of the tones which form them. Thus the beats of a whole tone, which, in low positions, are powerful and distinct, become less marked as we ascend in the scale, and in its highest portions prac- tically inaudible. Accordingly the beating-distance, which, for tones of medium pitch, we have roughly fixed at a Minor Third, must be supposed to contract somewhat for very high tones, while, for very low ones, it correspondingly expands. In consequence of this, a difference of relative pitch, which, in the lower part of the scale produces beats so slow as not perceptibly to interfere with smoothness of unison, may, in its higher region cause a harsh dissonance. We have here the reason why the ear is more sensi- tive to slight variations of pitch in high than in low notes, and why, therefore, greater accuracy in tuning 11—2 164 DISSOXANT OVERTOXES. [VIII. § 80. is essential to obtain a good unison effect from the former, than from the latter. The general partial-tone series consists of simple tones which, up to the seventh, are mutually out of beating-distance : above the seventh they close in rapidly upon each other. In the neighbourhood of 10, the interval between adjacent partial-tones is about a whole tone ; near 16, a semi-tone ; higher in the series they come to still closer quarters. These high partial-tones are therefore so situated as to produce harsh dissonances with each other. Where they are strongly developed in a clang, there will therefore be a certain inevitable roughness in its timbre. Tliis is the cause of the harsh quality of trumpet or trombone notes, and also of the shrill tremulous sounds sometimes observed in the human voice (p. 133). In fact we may regard all the portion of the partial-tone series above the eighth tone as contributing mere noise to the clang. Thus a noise may, conversely, be regarded as due to many simple tones within beating distance of each other. 80. It has been shown that, when two simple tones are simultaneously sustained, beats can arise directly between them only under one condition, viz. that the tones shall differ in pitch by less than a Minor Third, or thereabouts. When, however, the VIII. § 80.] DISSOXAFCE OF TWO CLANGS. 165 two co-existing sounds are no longer simple tones, but composite clangs, each consisting of a series of well developed partial-tones, the case becomes alto- gether different. Let us examine the state of things which then presents itself. The sounds of most musical instruments do not contain more than the first six partial-tones ; we will, therefore, assume this to be the case with the clangs before us. No beats can then arise between partial-tones of the same clang for the reason as- signed on p. 164. Dissonance due to beats will, however, be produced if a partial-tone belonging to one clang is within the specified distance of a partial-tone belonging to the other clang. Several pairs of tones may be thus situated, and, if so, each will contribute its share of roughness to the general effect. The intensity of the roughness due to any such pair will depend chiefly on the respective orders to which the beating partial-tones belong, and on the interval between them. The lowest partial-tones being the loudest, produce the most powerful beats, and half-tone beats are, in general, harsher than those of a whole tone. In determining the general effect of a combination of two clangs, we have to ascertain what pairs of partial-tones come within beating-distance, and to estimate the amount of roughness due to each pair. The sum of all these 166 CONSONANCE AND DISSONANCE. [VIII. § 81. roughnesses, if there are several such pairs, or the roughness of a single pair if there be but one, con- stitutes the dissonance of the combination. If there be no dissonance, the combination is described as a perfect concord. When dissonance is present, it vrill depend on its amount whether the combination is called an imperfect concord or a discord. The line separating the two must, of course, be somewhat arbitrarily drawn. 81. Let us examine the principal consonant intervals, in the manner above described, beginning with the octave, i ^^'-^ r ^l The minims here represent the fundamental- tones ; the crotchets above them corresponding over- tones. Those belonging to the higher clang are only written down as far as the third, since the fourth, fifth, and sixth have no corresponding tones of the lower clang disposable with which to form beating pairs. As long as the tuning is perfect each partial-tone of the higher clang coincides exactly with one belonging to the lower. No dissonance can consequently occur, and the combination is a per- yill. §81.] OCTAVK 167 feet coneord. But, suppose the higher C to be out of tune : each of its partial tones will be cor- respondingly too sharp or too flat, and three sets of beats will be heard between the partial-tones 2 — 1, 4 — 2, and 6 — 3. When the higher C is as much as a semi-tone wrong, the result is Efertl The pair 2 — 1 is of the mof^t importance, and gives in each case sixteen heats per second. The two others give respectively 32 and 48 beats per second. A semi-tone corresponds to about maximum roughness in the middle region of the scale, so that we have before us an exceedingly harsh discord. As the pitch of the higher note is gradually corrected, the rapidity of the beats diminishes, but the tuning must be extremely accurate to make them entirely vanish. If the note makes but one vibration per second too many, or too few, which corresponds to a difference in pitch of only about a thirtieth part of a icliole tone, the defect of tuning makes itself felt by three sets of beats, of 1, 2, and 3 per second respectively. The tunist must keep slightly altering the pitch 168 COIXCIDEXCE OF PARTIAL-TONES. [YIII. § 82. until he at length hits on that which completely extinguishes the beats. We saw in an earlier part of this inquiry (p. 63) that, when two sounds form with each other the interval of an octave, their vibra- tion-numbers must be in the ratio of 2:1. Long after it had been experimentally ascertained that rigorous comphance with this arithmetical condi- tion was essential for securing a perfectly smooth octave, the reason for this necessity remained entirely unknown, and nothing but the vaguest and most fanciful suggestions were offered to account for it — such as, for instance, that ''the human mind delights in simple numerical relations." This attempt at explanation overlooked the ob\dous fact that many people who knew nothing either about vibrations or the delights of simple numerical relations, could tell a perfect octave from an im- perfect one a great deal better than the majority of men of science. The true explanation, which it was left for Helmholtz to discover, lies in the fact, that only hy exactly satisfying the assigned nume- rical relation, can the ^9ar^2a7-fc>?2e5 of the higher clang he brought into exact coincidence ivith j^cirtial- tones of the lower, and thus cdl beats and conse- quent dissonance prevented, 82. No narrower interval than an octave can be found which gives an absolutely perfect concord. YIII. §. 82.] FIFTH, 169 The nearest approach to such a concord is the Fifth. ^ -172: Here we get two pairs of coincidences 3 — 2 and 6 — 4, but a certain roughness is caused by 3 of the higher clang being within beating-distance of both 4 and 5 of the lower clang. The tuning must be perfectly accurate, this interval being closely bounded by harsh discords. The result of an error of a semi- tone is as follows : — 11 o-^ SI E fe:; For every single vibration per second by which the higher clang is out, there will be two beats per second from the pair 3 — 2 with others of greater rapidity, but less intensity, from the higher pairs. The result, for the Fifth, is, therefore, that, however accurately tuned, it involves an appreciable rough- ness. It is true that since 4 and 5 are generally weak, and the beating intervals are whole tones. 170 FOURTH. [YIII. § 83. the rougKness will be very slight: still the trace of dissonance due to it prevents our classing the Fifth as an interval quite equally smooth with the octave. 83. For the Fourth we have \t S^ The amount of dissonance is greater than in the case of the Fifth, since 3 and 2 are usually hotli powerful tones, and produce therefore louder beats than those of 4 — 3 and 5 — 3. There are, in addi- tion, the beating pairs 6 — 4 and 6 — 5. Moreover the first pair of coincident or partial tones, 4 — 3, are, in general, weaher than the beating pair below them, 3 — 2. The Fourth is bounded only on one side by a harsh discord. If its upper clang is half a note too sharp, we have the interval C — jPjf, which is treated in the last figure but one. Slight flattening of the F will set the pair 4 — 3 beating slowly ; the disappear- ance of their beats thus secures the accurate tuning of the interval. On the other hand, lowering F weakens the beats of 3 — 2, by widening the dis- tance between those tones, and, therefore, tends ,to lessen the whole amount of roughness. These yill. §§ 84, 85.] MAJOR THIRD AND MAJOR SIXTH. 171 considerations go far to explain tlie fact that a long dispute runs through the history of music, as to whether the Fourth ought to be treated as a concord or as a discord. The decision ultimately arrived at in favour of the first of these alternatives was perhaps, as Helmholtz suggests, due more to the fact that the Fourth is the inversion of the Fifth, than to the inherent smoothness of the former interval. 84. Next come the intervals of the Major Third and Major Sixth, which shall be taken together, as they are very nearly equally consonant. 4.a^.-~,^._fl 3_T_ 4.^ — ^2— -3 et The dissonance due to the pair 3 — 2, separated by a tone in the Sixth, is perhaps, about equal to that of the weaker pair 4 — 3, which are only a semi-tone apart, in the Third. The definition of these intervals depending, as it does (in both) on the ffth tone of the lower clang, will in general be but feebly marked. 85. The last remaining intervals, less than an octave, which rank as concords, are those of the Minor Third and Minor Sixth. Each contains strong elements of dissonance; in 172 MIXOR THIRD AXD MIXOR SIXTH. [YIII. § 86. fact, we are here near tlie boundary line between con- cord and discord. As regards sharpness of definition, _,i.^_J^ ^gi-3- ^-r — P#-2- :^- 5221 -ft©-i— ■ the tones 6 and 5, on which it depends in the first of the two intervals, are, in the sounds of most instruments, weak or even entirely absent, while for the second interv^al the series of partial-tones must be extended as far as the 8th of the lower clang in order to reach the first coincident pair. Accordingly the Minor Sixth can hardly be said to be defined at all, for clangs of ordinary quality, by coincidence of partial-tones. Its powerful beating pair 3 — 2, separated by the interval of greatest dissonance, a semi-tone, makes it the roughest of all the concords. On the pianoforte, and other instru- ments with fix:ed tones, the same notes (CA\>) ivhicli o^epresent the Minor Sixth have also to do duty as one of the harshest discords, the Sharp Fifth, (CGi), The extremely defective consonance of the Minor Sixth could hardly be more conclusively shown than by the fact just mentioned. 86. As regards the dissonant intervals of the scale, we have, in addition to those incidentally examined above, the semi-tone, tone, and ]\Iinor VIII. § 87.] DISSONANT INTERVALS. 173 Seventh. The first two need not be examined, since obviously each pair of corresponding overtones are brought v^ithin the same beating intervals as the two fundamentals. The dissonance resulting is, of course, harsher for the half than for the whole tone. The Minor Seventh is constituted thus : m $=zi^ ■l2p-' BI It is the mildest of the discords, in fact, in actual smoothness it decidedly surpasses the Minor Sixth. 87. In order that the reader may see at a glance the whole result of this somewhat laborious discussion, we subjoin a graphical representation of the amount of dissonance contained in the several intervals of the scale. The figure is taken, with some slight alterations, from that given at p. 519 of Helmholtz's Work. The intervals, reckoned from (7, are denoted by distances measured along the horizontal straight line. The dissonance for each interval is represented by the vertical distance of the curved line from the corresponding point on the horizontal line. The cal- culations on which the curve is based were made by Helmholtz for two constituent clangs of the quahty of the violin. For pianoforte sounds the form of 174 PICTURE OF INTERVALS. [VIII. § ^^. the curve would be slightly different ; for those of stopped organ-pipes, &c. very different indeed. :Fig . 57 The figure indicates the sharpness of definition of an interval by the steepness with which the curve ascends in its vicinity. If we regarded the outline as that of a mountain-chain, the discords would be represented by peaks, and the concords by passes. The lowness and narrowness of a particular pass would measure the smoothness and definition of the corresponding musical interval. 88. The theory of musical consonance and dissonance, our examination of which is now con- cluded, necessarily leads us to regard the distinc- tions between different concords laid down by theo- retical musicians as not in themselves absolute, but dependent on the quality of the sounds experimented upon. The results we have arrived at are generally true for sounds containing the first six partial-tones, but they will not apply, without modification, to clangs differently constituted. To take a case or YIII. § 88.] QUALITY AND CONSONANCE. 175 two in point. Suppose, for instance, we are dealing with sounds such as those of stopped organ-pipes which contain only odd partial-tones (p. 121). It is at once clear from p. 169 that the interval of the Sharp Fourth C — F^, which owes its dissonant charac- ter to the beating pairs 3 — 2, 4 — 3, and 6 — 4, will become something quite different when the disso- nance due to all these pairs disappears, as it must do, since each of them contains at least one partial-tone of an even order. The Minor Sixth would also gain in smoothness in such a timbre, by the removal of the loudly discordant pair 3 — 2. Helmholtz has examined the case of a hautbois taking one note of an interval and a clarionet the other, and shown that some concords sound best when the former instrument plays the upper note and the latter the lower, while with others the reverse is the case. The hautbois produces the un- interrupted series of partial-tones, the clarionet only its odd members. Reference to p. 171 shows that, in the case of the Major Third, we can only get rid of the dissonant pair 4 — 3 by assigning the higher note to the hautbois. The Fourth, on the contrary, will be seen, by p. 170, to sound smoothest when the clarionet is above the hautbois, since by this arrangement we divorce the quarrelsome couple 3 — 2, whose bicker- ings will, in the opposite position, continue to be 176 CASE OF SIMPLE TOXES. [YIII. §§ 89, 90. heard. These conclusions, which experiment con- firms, are, I believe, in advance of any obtained empirically by musical theorists. Corresponding rules might easily be elicited for other instruments. 89. It is possible to draw from the general theory of consonance and dissonance an inference which seems, at first sight, fatal to the truth of the theory itself. " If," it may be said, ^^ the difference between a consonant and a dissonant interval depends entirely on the behaviour towards each other of certain pairs of overtones ; then, in the case of sounds like those of large stopped flue- pipes, luhere there are no overtones at all, the dis- tinction between concords and discords ought en- tirely to disappear, and the mterval of a Seventh, for instance, to sound just as smooth as that of an octave. As this is notoriously not the fact, the theory cannot be true." In order to meet this objection, it will be necessary first to acquaint the reader with certain known expe- rimental facts which Helmholtz has dragged out of the obscurity in which they had lain for fully a century, and forced to deliver thek testimony in favour of his theory. 90. Let two tuning-forks of different pitch mounted on their respective resonance-boxes, and therefore producing simple tones, be thrown into YIII. § 90.] COMBINATION-TONHS. 177 ])owerfal vibration by the use of a resined fiddle- bow. With adequate attention, it is possible to recognize, in addition to the tones of the forks them- selves, certain new sounds, which usually differ in pitch from hoth the former ones. These tones, called, from the manner of their production, comhina- t ion-tones, fall into two classes, with only one of which, discovered in 1740 by a German organist named Sorge, we need here concern ourselves. It consists of a series of tones called combination- tones of \h.Q first, second, third, &g., orders, of which the first is of the most importance, as it can be heard without difficulty. Its pitch is determined by the following law. The combination-tone of the first order of two simple primary-tones has for its vihra- tion-number the difierence between the respective mbration-numbers of the primaries. Thus, e.g., if the two primaries make 200 and 300 vibrations per second, and therefore form a Fifth with each other, the first combination-tone will make 100 vibrations per second, and, accordingly, lie exactly one octave below the graver of the two primary tones. In this manner we can determine the com- bination-tones of the first order for pairs of simple primaries forming any given interval with each other. The following table, copied from Hehnholtz's work, T. 12 178 COMBIXATION-TOXES. [YIII. § 90. shows the results for all the consonant intervals not exceeding one octave. Interval. Vibration-ratio. Difference. Depth of the Combination-tone i below the graver primary. j 1 Octave 1 :-2 Unison Fifth 2 : 3 Octave Fourth 3 : 4 Twelfth Major Third 4 : 5 Two Octaves Minor Third 5 : 6 Two Octaves t Major Third ; Major Sixth 3 : 2 Fifth Minor Sixth 5 : 8 3 Major Sixth In musical notation the same thing stands thus, the j^i^ii^aries being denoted by minims, and the combination-tones by crotchets. ='iliPiliiii' ^: ei n^i :2«: Combination-tones are produced with remark- able distinctness by the harmonium. If the prima- ries shown in the treble stave are played on that instrument while the pressure of air in the bellows is vigorously sustained, the corresponding combina- nation-tones of the first order, written in the bass, come out Avith unmistakeable clearness. They are in fact much better heard thus than from tuning-forks. yill. §91.] INTEXSITY OF C02fBmATI0N--T0NES. 179 Combination-tones of the second order may be treated as if tliey were first-order tones produced between one or otlier of tbe primaries and the com- bination-tone of the first order. Similarly we may regard each combination-tone of the third order as due to a second-order tone, paired either with one of the primaries, with the first-order tone or with its own fellow of the second order. Successive subtraction will therefore enable us to determine the vibration- number of a combination-tone of any order from the vibration numbers of the two primaries. Combination-tones grow rapidly feebler as their order becomes higher. Those of the first order are usually distinct. enough, and those of the second to be heard with a little trouble. The third order is only recognizable when entire stillness is secured, and the greatest attention paid. It is a moot point whether the fourth- order tones can be heard at all. 91. We can now show that the existence of combination-tones prevents intervals formed by two simple tones from altogether lacking the characteristic differences of consonance and dissonance, though those differences are far less marked than in the case of composite sounds. To begin with the octave. Let us suppose that we have two sunple tones form- ing nearly this interval, but that the higher of them 12—2 180 BEATS OF COMBIXATIOX-TOXES. [VIII. § 91. is a little sharp, so that the octave is not strictly ill tune, is in fact slightly impure. Let the lower tone make 100, the higher 201, vibrations per second. They will give rise to a combination-tone making 101 vibrations ^qv second (p. 177), and this mth the lower primary will produce one heat per second. If the higher primaiy had been flat, instead of sharp, making, say, 199 vibrations per second, we should have had 99, as combination-tone, giving rise, with 100, to beats of the same rapidity as before. These beats cannot be got rid of except by making the vibration-ratio exactly 1:2, i. e. the octave perfectly pure. The roughness must increase both when the interval widens and when it contracts, so that* the octave, in simple tones, is a well-defined concord bounded on either side by decided discords. This result may be easily verified experimentally by taking two tuniiio'-forks forminof an octave with each other, and throwing the interval slightly out of tune by causing a pellet of wax to adhere to a prong of one of them. On exciting the forks the beats will be distinctly heard. The octave is the only interval which is defined by the beats of a combination-tone of the first order with one of the primary tones. For the next smoothest concord, that of the Fifth, we are obliged VIII. § 92.] INFLUENCE OF COMBINATION-TONES. 181 to have recourse to the second order. Thus, pro- ceeding as in the case of the octave, we have Primaries 200 301 C. T. of 1st order ^101 C. Ts. of 2nd order 99 200 No. of beats per sec. 2 The Fifth is, thus, a fairly well-defined consonance, though decidedly less sharply bounded than the oc- tave, owing to the feebleness of the C. T. of 2nd order. For the Fourth we have Primaries 300 401 C. T. of 1st order 101 C. Ts. of 2nd order 199 300 C. Ts. of 3rd order 101 202 98 No. of beats per sec. 3 The 3rd- order tone being excessively weak, the interval of a Fourth can scarcely be said to be de- fined at all. Still less can the remaining consonant intervals of the scale, by the evanescent beats of still higher orders of combination-tones. 92. With simple-tones, then, the case stands thus. The intervals of a Second and a Major-Seventh are palpably dissonant, owing to the beats of the primaries, in the former, and of a first-order combi- nation-tone with a primary, in the latter. There is a certain amount of dissonance in intervals slightly 182 CONSONANCE AND QUALITY. [YIII. § 92. narrower or slightly wider than a Fifth, but of a feebler kind than in the case of the octave, inasmuch as it is due to only a second- order combination-tone. Whatever dissonance may exist near the Fourth is practically imperceptible. All other intervals are free from dissonance. Accordingly all intervals from the Minor Third nearly up to the Fifth, and from a little above the Fifth up to the Major Seventh, ought to sound equally smooth. This conclusion is probably very inconsistent with the views of musical theorists, who regard concord and discord as entirely indej)endent of quality, but it is strictly borne out by experiment. With the tones of tuning-forks the intervals lying between the Minor and Major Thirds, and between the Mmor and Major Sixths, though sounding somewhat strange, are entirely free from roughness, and, therefore, cannot be described as dissonant. As a further verification of this fact, Helmholtz advises such of his readers as have access to an organ to try the effect of playing alternately the smoothest concords and the most extreme dis- cords which the musical scale contains, on stops yield- ing only simple-tones, such, e.g., as the flute, or stopped diapason. The vivid contrasts which such a proceeding calls out on instruments of bright timhre, like the pianoforte and harmonium, or the more bril- liant stops of the organ, such as principal, hautbois. YIII. § 93.] COMBINATION-TONES OF OVERTONES. 183 trumpet, &c., are here blurred and ejffaced, and every- thing sounds dull and inanhnate, hi consequence. Nothmg can show more decisively than such an experiment that the presence of over-tones confers on music its most characteristic charms. Thus the remark put into the mouth of a sup- posed objector in § 89 turns out to be no objection whatever to Helmholtz's theory of consonance and dissonance, but, so far as it represents actual facts, to be valid against the prevalent views of musical theorists. 93. It may we well to advert briefly, in this place, to a point connected with combination-tones which may otherwise occur as a difficulty to the reader's mind. When two clangs coexist, combina- tion-tones are produced between every pair which can be formed of a tone from one clano^ with a tone from the other. These intrusive tones will usually be very numerous, and, for aught that appears, may interfere with those originally present, to such an extent as to render useless a theory based on the presence of partial-tones only. Helmholtz has re- moved any such apprehension, by showing that, in general, dissonance due to combination-tones produced between overtones, never exists except ivJiere it is already present by virtue of direct action among the overtones themselves. Thus the only effect attri- 184 C02IBIN-ATI0N-T0NES OF OVERTONES. [YIIL § 93. butable to this source is a somewhat increased roughness in all intervals except absolutely perfect concords. No modifications, therefore, have to be introduced, on this score, in the conclusions of §§ 81—86. CHAPTER IX. ON CONSONANT TEIADS. 94. In tlie ensuing portion of this enquiry we shall have to make more frequent use than hitherto of vibration-fractions. It may, therefore, be well to explain the rules for their employment, in order that the student may acquire some facility in hand- ling them. The vibration-fraction of an assigned interval expresses the ratio of the numbers of vibrations performed in the same time by the two notes which form the interval. The particular length of time chosen is a matter of absolute in- difference. The upper note of an octave, for instance, vibrates twice as often as the lower does in any time we choose to select, be it an hour, a minute, a second, or a part of a second. It is often convenient to determine the vibration- fraction of an interval from the vibration-numbers of its constituent notes : in such a case we choose one second as our time of comparison, and in this way vibration-fractions were 18G MEAXIXG OF VIBRATIOX-FEACTIOXS. [IX. § 95. defined in § 34 : any otlier standard is however equally legitimate, though, in general, less convenient. To illustrate these remarks on a particular case, the vibration-fraction ^ indicates that, while the lower of two notes forming a Major Third makes four vibrations, the higher of them makes ^xe. There- fore, while the lower makes one vibration, the hio^her makes |ths of a vibration, or 1^ vibrations. Con- versely, while the liigher note makes one vibration, the lower makes |-ths of a vibration. The same reason- ing being equally applicable to other cases, it follows that any fraction greater than unity denotes the number of vibrations, and fractions of a vibration, made by the higher of two notes forming a certain interval, while the lower note is making a single vibration. Similarly, any fraction less than unity indicates the proportion of a whole vibration per- formed by the lower note, while the upper is making one complete vibration. The rules for addinof and subtracting^ intervals shall next be laid down. 95. Suppose that, starting from a given note, a second note, a Fifth above it, is sounded, and then a third note, a Major Third above the second. What will be the vibration-fraction of the interval formed by the first and third notes, i. e. of the sum of a Fifth and a Major Third ? We will, for short- IX. § 96.] ADDITION OF IXTEMVALS, 187 ness, call the tliree notes (1), (2), (3) in order of ascending pitch. The vibration-fractions being, for (l) — (2), f, and, for 2—3, f, we proceed thus : While (2) makes 4 vibrations, (3) makes 5 vibrations. Therefore, while (2) makes 1 vibration, (3) makes -| vibrations. Therefore, while (2) makes 3 vibrations, (3) makes 3xf vibrations. But while (2) makes 3 vibrations, (1) makes 2 vibrations. Therefore, while (1) makes 2 vibrations, (3) makes 3x| vibrations. Therefore, while (1) makes 1 vibration, (3) makes |x| vibrations. Our result, then, is the two vibration-numhers multiplied together. The reasoning is perfectly ge- neral, and gives us the following rule. To fold the vihration-fractioji for the sum of iivo intervals, midtiply their se2^arate vihration- fr actions together. 96. Next, take the opposite case. Let (2) be a Major Third above (1), and (3) a Fifth above (l), and let the vibration-fraction for the interval (2) — (3) be required. While (1) makes 4-vibrations, (2) makes 5 vibrations. Therefore, while (1) makes 1 vibration, (2) makes | vibrations. But, while (1) makes 2 vibrations, (3) makes 3 vibrations. Therefore, while (1) makes 1 vibration, (3) makes f vibrations. Hence, while (2) makes | vibrations, (3) makes f vibrations. Therefore, while (2) makes a of a vibration, (3) makes lx% of a vibration. Therefore, while (2) makes 1 vibration, (3) makes |x| vibrations. 188 SUBTRACTIOX OF INTERVALS. [IX. § 97. The result here is the quotient resulting from the division of the larger vibration-fraction hy the smaller: hence we have this general rule. To find the vibration fraction for the difference of two ■ intervals, divide the vibration fraction of the wider by that of the narroiver interval. Thus multiplication and division of vibration- fractions correspond to addition and subtraction of intervals^. 97. One of the smiplest cases of our second rule occurs when an interval has to be inverted. The * inversion ' of any assigned interval narrower than an octave is the difference between it and an octave, i, e. the interval which remains after the first has been subtracted from an octave. Thus to find the vibration-number for the inversion of the Minor Third we merely have to divide 2 by |-, or in other words invert the vibration-fraction of the interval and multiply by 2. This applies to all cases. In the particular example selected, the result is -| ; the inversion of the Minor Third is therefore the Major Sixth. The relation between an interval and its inversion is obviously mutual, so that each may be ^ By simply reducing tlie numerical results, obtained in §§ 95, 96, the student will establish the following propositions : ' A Major Third added to a Fifth produces a Major Seventh.' 'A Major Third subtracted from a Fifth leaves a Minor Third.' IX. §§ 98, 99.] INVERSION OF INTERVALS. 189 described as the inversion of the other. Accordingly the inversion of the Major Sixth is the Minor Third. The following, table shows the three pairs of con- sonant intervals narrower than an octave, which stand to each other in the mutual relation of inver- sions. Minor Third (|)— Major Sixth (|) Major Thh^d (|)— Minor Sixth (f) Fourth (I)— Fifth (f ) The student is advised to verify, by the method of p. 188, the fact that each of these intervals is the inversion of that placed by its side. 98. A combination of musical sounds of different pitch is called a * chord.' Hitherto we have con- sidered only chords of two notes, or 'binary' chords. We now go on to chords of three notes, or, as they are usually called, 'triads.' A binary chord is, of course, consonant if its two notes form a consonant interval. A triad contains three intervals, one be- tween its extreme notes, and onie between the middle note and each of the other two. In order that the chord may be free from dissonance, those intervals must all three be concords. 99. We may, then, search for consonant triads in the followinof manner. Havino; selected the lowest of the three notes at pleasure, choose two others, each of which forms ivith the bottom note a consonant 190 COKSOXAKT TRIADS. [IX. § 99. interval. Next, examine whether the interval formed hy the last chosen notes loith each other is also a concord. If so, the triad itself is consonant. In order to determhie all the consonant triads within an octave above the fixed bottom note, we must assign to the middle and to23 notes every possible consonant position with respect to the bottom note, and reject all such relative positions as give rise to dissonant intervals between those notes themselves. The re- maining positions will constitute all the consonant triads which have for their lowest note that ori- ginally selected. The intervals at our disposal are, for the middle note, from the Mmor Thkd to the Minor Sixth, and, for the upper note, from the Major Third to the Major Sixth. In the annexed table ^ the possible positions of the middle note with respect to the bottom note, are shown in the left-hand vertical column, the name of each interval being accompanied by its vibration-fraction. The possible positions of the top note are similarly shown in the highest horizontal column. Each space common to a horizontal and a vertical column con- tains the vibration-fraction of the interval formed between the simultaneous positions of the middle and upper notes named at their extremities. Where ^ This table is copied with slight modifications from Helm- holtz's work. IX. § 100.] CONSONANT TRIADS. 191 these intervals are dissonant, their vibration-fractions are enclosed in square brackets. When they are concords the name of the interval is, in each case, appended. Major Third Fourth Fifth 1 Minor Sixth Major Sixtli 1 % Minor Third 6 [S] m Major Third 4 Fourth [^] Major Third 5 4 [5] 1 Minor Third ^^^ Fourth Fourth 1 1 1 1 m Minor Third 1 Major Third Fifth [ff] m Minor Sixth 8 a [Ml 100. The followino', then, are all the cases : ■ Middle Note. 1 Upper Xote. Minor Third Fifth, or Minor Sixth Major Third Fifth, or Major Sixth Fourth Minor Sixth, or Major Sixth 192 MAJOR AND MINOR GROUPS. [TX. § 101. or, in musical notation, tel^i^|E^|3ii^ --9-& They give ns two groups of three iiv.ijor, and three minor, triads, which may be arranged thus : 'Fifth. ^^^^ iMajor Third. ^^ (Minor Third Minor Sixth. TMajor Sixth. ^'^ iFourth. ^""^ iMmor Third. ^H Major Third. ^^^ iFourth. 101. Instead of defininof our six consonant triads, as we have done, by the mtervals formed by their middle and top notes wdth the bottom note, Ave may define them by the intervals separating the middle from the bottom note, and the top from the middle note. In order to make this change we have, in each case, a process of subtraction of intervals to perform. Thus the difference between a Fifth and a Major Third is f x f , i. e. f, or a Minor Third. Proceeding in this way, we find that the top and middle notes are separated by the followuig in- tervals : 1 (^) {-) 03) b) ]Miuor Tliird Fourth Major TLkd Major Tliird Fourth Minor Third IX. § 101.] INVERSION OF TRIADS. 193 Hence we may write our two groups as follows : («'){ Minor Third. rFourthc f Major Third. Major Third. ^ ^ (Minor Third. ^"^ ^ iFourth. . rMajor Third. rFourth. f Minor Third. ^""^ {Minor Third. ^ ^Major Third. ^'^ ^ iFourth. It will now be easy to show that the triads of each group are very closely connected together. Take (ci), and let us form another triad from it, by causing its bottom note to ascend one octave, the other two remaining where they were. The middle will then become the bottom note, the top the middle note, and the octave of the former bottom note the top note. Hence the lower interval of the new triad will be the upper interval of the old one, i.e. a Minor Third. The upper interval of the new triad will necessarily be the inversion of the interval which separated the extreme notes of the old triad. This interval is a Fifth [see (a), p. 192], and its inversion, by the table on p. 189, is a Fourth. Hence the new triad is <^_. l, . w which is identical with ib'), IMmor Third,J If we modify {Ij) in the same way, the new interval is the inversion of the Minor Sixth, i. e. the Major Third, and the resulting triad, viz. Major Third) ... Fo fh i ^^ identical with {c). This triad, when T. 13 19i INVERSION OF TRIADS, [IX. § 102. similarly treated, brings us back to (a), and the cycle of changes is complete. By an extension of the word 'inversion/ it is usual to call the triads {U) and [c) the first and second inversions of the triad Exactly similar relations hold between the mem- bers of the second group of triads : (/3') and {y) are, accordingly, called the first and second inver- sions of the triad (a). The proof is exactly like that just given, and will be easily supplied by the reader. 102. If we choose C as the bottom note of [a) and (a'), the major and minor groups will be ex- pressed in musical notation by E = iE£ and ^tbHE=t C2_ («') {^') (C') (a) (/3') W) They may also be defined in the language of Thorough Bass, which refers every chord to its lowest note, in accordance with the mode adopted in [a), {h), (c); (a), {^), (y). Thus the triads (a), (6'), {c) would be indicated by the figures |, I, I respectively, and so ivould the triads (a) {/B') (y) ; the differences be- tween Minor and Major Thirds and Sixths being left to be indicated by the key-signature. The positions (a) and (a ) are regarded as the fundamental ones of each group, (h') {c) and (/3') (y') IX. §103.] COMMON-CHORDS. ' 1^)5 being treated as derived from tliem respectively by inversion. 103. The fundamental triads bear tlie name of their lowest notes, thus (a) and (a') are called respectively the major and minor common^chords of a The remaining members of each group are not named after their own lowest note, but after that of their fundamental inversion ; thus (6^) (c) and (^') (y) are respectively major and minor common- chords of G in their first and second inversions^ The reason of this, as far as the major group is concerned, follows, directly from Helmholtz's theory of consonance and dissonance. The notes of the triads (a), (6'), {c) are all coincident with indivi- dual overtones of a clang whose fundamental-tone is the low (7, ^ ^=— { for (a') and (6'), and the octave above that note for (c ) : hence they may be regarded as forming a part of the clang of a (7-sound, and therefore each triad may be appropriately called by its name. With the Minor triads this is not so com- pletely true, because the E ^^ in (a ), (/8'), (y') is not coincident with an overtone of C, The other two notes, however, are ia each case leading partial-tones of the clang of (7, and therefore these triads belong at any rate more to C than to any other note. 13—2 196 MAJOR AND MINOR EFFECTS. [IX. § 104. Common-cliords of more than three constituent sounds can only be formed by adding to the con- sonant triads notes which are exact octaves above or below those of the triads. 104. The marked distinction existing, for every musical ear, between the bright open character of major, and the gloomy veiled effect of minor chords, is attributed by Helmholtz to the different way in which combination-tones enter in the two cases. The positions of the first-order combination-tones, for each of the six consonant triads, are shown in crotchets in the appended stave, the primaries being indicated by minims. Each interval gives rise to its own combination-tone, but, in the cases of the funda- mental position and second inversion of the (7-Major triad, two combination-tones happen to coincide. The reader will at once notice that in the major group no note extraneous to the harmony is brought in by the combination-tones. In the minor group this is no longer the case. The fundamental position, and the first inversion, of the triad, both bring in an A \>, which is foreign to the harmony, and the second inversion involves an additional extraneous note, B k The position of these adventitious sounds is not such as to produce dissonance^ for which they are too far apart from each other and from the notes of the triad; but they cloud the transparency of the har- IX. § 104.] MAJOR AND MINOR EFFECTS. 197 mony, and so give rise to the effects cliaracteristic of the minor mode. :3=2^=-3! ^ y — J—& (SJ- — ig & .i^E I The unsatisfying character of Minor, compared with Major, triads, comes out with peculiar distinct- ness on the harmonium ; as indeed, from the powerful combination-tones of that instrument, we should naturally have anticipated. , CHAPTER X. ON PUEE INTONATION AND TEMPEKAMENT. 105. The vibration-fractions of the intervals formed by tbe successive notes of the Major scale ivith the tonic, are, including the octave of the tonic, these : ^ 5. 4 3. 8. 5. 15 9 8» 4' 3' 2? 5' 3' 8 ' "" The intervals hetiveen successive notes of the scale are determined by dividing each of these fractions by that which precedes it. Thus the consecutive intervals of the Major scale come out as follows : CDEFGABC P_ 10 16 0. 10 0. lA 8 9 T^B "9 8 15 Only three different intervals are obtained. | is .slightly wider than ^^ ; ^ decidedly narrower than the other two. f and ^^ are called whole tones, ^ a half-tone or semi-tone, though, strictly speak- ing, two intervals of tliis width added together somewhat exceed the greater of the two whole tones ; since xf >< il ^^ IH i^ ^^ f ^^ ^^^ ^^^^^ ^^ 2048 to 2025. X. § 105.] REQUISITES FOR PURE INTONATION 199 Suppose we liad a keyed instrument containing a number of octaves, each divided into seven notes, forming the ordinary scale as above : any music could be played on it which did not introduce notes foreign to the key of C Major. But now, suppose we wanted to be able to play in another Major key as well as in that of (7, for instance G. It would be necessary for this purpose to introduce two new notes in every octave of the key-board. If G is the new tonic, A will not serve as the second of its scale, because the step between tonic and second is, not ^^, but f. Hence we must have a fresh note lying between A and B, Further, F will not do for the seventh of the scale of G, as it is separated from G by |-, instead of -^f. This necessitates a second additional note lying between F and G. If we take, as our original octave, that from middle- (7 upwards, we have the following vibration-numbers : CDEFGABC 264 297 330 352 396 440 495 528 The new notes, being respectively f above, and -^ heloiv G, have for their vibration-numbers f x 396 and If X 396, i.e. 445|- and 371 J. The other notes of the scale of G Major can be supplied from that of C Major. Hence these two scales are closely con- nected with each other. Another key nearly related 200 REQUISITES FOR PURE INTONATION. [X. § lOG. to the key of C is that of i^. Its Fourth is | x 352, or 46 9 J, which falls between A and B, Its Major Sixth is |- X 352, or 586|, which is clearly not an exact octave of any note between C and C\ The cor- responding note in our octave, found by division by 2, is 293-J-, which comes between C and D. Thus, two more new notes in the octave must be introduced, to make the key of i^ major attainable. 106. In order that the reader may see, at a glance, the variety of sounds which are requisite to supply complete Major scales for the seven keys of (7, D, E, F, G, A and B, the vibration- numbers for all the notes of these scales are cal- culated out and exhibited in the following table. Beducing those notes which lie beyond the Tonic Second Major Third Fourth Fifth Major Sixth Major Seventh C, 264 297 330 352 396 440 495 D, 297 334i 37U 396 445i 495 5561 E, 330 37U 412i 440 495 550 6181 F, 352 396 440 469J- 528 586f 660 G, 396 4451 495 528 594 660 742* A, 440 495 550 586f 660 733J 825 B, 495 5561 6181 660 7421 825 928^ X § 107.] REQUISITES FOR PURE INTON-ATION: 201 octave, by dividing them by 2, and arranging the results in order of magnitude, we have twelve notes foreign to the scale of C Major, the positions of which, with reference to the notes of that scale, are as follows : (7, 275, 278t^, 293J-, D, 309|, E, 3341 p^ 366|, 371^, G, 4121 4171, A, 445i, 464^1 , 469^ B, 107. If it is desired to be able to play in the Minor mode of each of the seven keys, as well as in the Major, additional notes will be called for. Each scale must contain three Minor intervals, viz. Minor Third, Minor Sixth, and Minor Seventh. The following subsidiary table exhibits the vibration- numbers of the sounds forming these intervals with the successive key-notes. Tonic Minor Third Minor Sixth Minor Seventh C, 264 316i 4221 469J D, 297 3561 475i 528 E, 330 396 528 586f F, 352 422f 5m 625.^ G, 396 4751 6334 704 A, 440 528 704 782f B, 495 594 792 880 202 REQUISITES FOR PURE INTONATIOF. [X. § 108. Reducing these to one octave, as before, we find seven notes not included in the previous list, oc- cupying the following positions : C, 281f, D, 312|, 316f, E, F, 35Gf, 3911 G, 422f, A, 4751 B. 108. Hence, to play perfectly in tune in both Major and Minor modes of the seven keys (7, i), E, F, G, A, B, it is necessary to have a key-board with twenty-six notes in every octave. This number, large as it is, by no means includes all necessary notes. Modern music is written in sharp and^a^ keys, i. e. in such whose tonics are not coincident with any one of the notes CDE...B. Moreover, the sharp and flat key-notes are different from each other. Thus G^, being a Major Third above E, is, as the first table shows, 412J ; while ^b is seen, by the second table, to be 422f , which is a somewhat sharper note. As the seven keys which have been already examined require 26 notes in the octave, we may anticipate that the ten additional sharp and flat keys will bring in a still larger number. It is needless to institute a detailed enquiry into these scales, but, after what he has already seen, the student will feel no surprise when he learns that a competent authority^, who has examined the ^ Mr A. J. Ellis, Proceedings of the Royal Society, Yol. xm. p. 98. X. § lOD.] DEFECTS OF KEYED INSTRUMENTS. 203 'subject most minutely in reference both to melodic a.nd barmonic requisites, fixes 72 notes in the oc- tave as tbe number essential to theoretically com- plete command over all the keys used in modern ^music« 109. Without, however, assuming this result, the facts we have already ourselves established are amply sufficient to show how serious are the imper- fections of tune which inevitably beset instruments with fixed tones, such as the pianoforte, harmo- ,nium and organ, containing only tivelve notes in each octave. Pure intonation in the ^ natural ' keys alone, i. e. those whose tonics are white notes on the 'board, demands, as has been seen, more than twice this number of available sounds ; and many more still, if the keys with tonics on the black notes are to be included. Perfect tuning in all the keys being entirely out of the question, a compromise of some kind is the only possible course. Thus we may tune a single key, say C, perfectly ; in which case most of the other keys will be so out of tune as to be unbearable. Or again, we may distribute the errors over certain often-used keys, and accumu- late them in others which are of less frequent oc- currence. Expedients of this kind are described as modes of 'tempering,' and the system adopted in tuning 204 'TEMPERING' AND 'TEMPERAMENT: [X. § 110. any particular instrument is called its 'temperament.' A vast number of different methods of tempering have been proposed and tried during the history of the organ and pianoforte. 110. That which has at last been almost univer- sally adopted is the system of equal temperament. It consists in dividing each octave into twelve pre- cisely equal intervals. Each of these intervals is called a semi-tone, and any two of them together a whole tone. The octave of which C, 264, is the lowest note, will contain, on the equal temperament system, the following sounds. The vibration-numbers are given true to the nearest integer. When a note is slightly sharper than that so indicated, this is shown by the sign + attached to the vibration-number in ques- tion ; when slightly flatter, by the sign — . For the sake of comparison, the perfect intervals of the same scale are written below the tempered ones. C, Cjj, D, D#, E, F, fJ, G, G|. a, aJ, B 264, 280-, 296+, 3U-, 333-, 352+, 373+, 39o+, 419 + ,444-, 470+, 498+ C, D, Eb, E, F, G, Ab, A, Bb, B 264, 297, 317-, 330, 352, 396, 422+, 440, 469+, 495 It is clear that the regions of the tempered scale where the tuning is the most imperfect, are in the neighbourhood of the Thirds and Sixths. E and A X. § 1 1 1.] EQUAL TEMPERAMENT. 205 are nearly three vibrations per second too sharp. The Fourth and Fifth are less out of tune, in fact only wrong by a fraction of a vibration per second. 111. The intervals of the tempered scale are so nearly equal to those of the perfect scale, that, when the notes of the former are sounded successively, it re- quires a delicate ear to recognize the defective charac- ter of the tuning. When, however, more than one note is heard at a time, the case becomes quite different. We saw in Chapter viii. how rigorously accurate the tuning of a consonant interval must be, to secure the greatest smoothness of which it is capable. It was also shown that such intervals are generally very closely bounded by harsh discords. Now since, in the system of equal temperament, no interval except that of the octave is accurately in tune, it follows that every representative of a concord, in its scale, must be less smooth than it would be were the tuning perfect. One of the greatest charms of music, and especially of modern music, lies in the vivid contrast presented by consonant and dissonant chords in close juxtaposition. Temperament, by im- pairing, even though but sHghtly, the perfection of the concords, necessarily somewhat weakens this contrast, and takes the edge off the musical pleasure which, in the hands of a great composer, it is capa- ble of giving us. A fact already once adverted to 206 EFFECT OF TE2IPERAMEXT, [X. § 112. (p. 172) may be again adduced here, as illustrating the effect of temperament in blurring distinctions of con- sonance and dissonance, viz. that on the key-board of the pianoforte, the same two notes which represent C A^, which is a concord, though not a very smooth one, also appear in (7 - (r^, which is a decided discord. A reference to § 108 will show that, with perfect tuning, Gf, and A^ are different notes havhig vibra- tion-numbers in the proportion of 412-J- to 422|-. One of the readiest ways of recognizmg the de- fective character of equal temperament tunmg is, first, allow a few accurate voices to sing a series of sustained chords in three or four parts, without accompaniment, and then, after noticing the effect, to let them repeat the phrase while the parts are at the same time played on the pianoforte. The sour character of the concords of the accompanying instrument w^ill be at once decisively manifested. Voices are able to smg perfect intervals, and their clear transparent concords contrast with the duller substitutes provided by the pianoforte in a way obvious to every moderately acute ear. 112. Since the voice is endowed with the power of producing all possible shades of pitch within its compass, and thus of smging absolutely pure inter- vals, it is clear that we ought to make the most of this great gift, and especially in the case of those X. § 112.] VOCAL IXTOXATIOX, 207 persons who are to be public singers, allow, during the years of preparation, no contact save with the purest examples of intonation. Unfortunately the practice of most singing-masters, is the very reverse of this. The pupil is systematically accompanied, during vocal practice, on the pianoforte, and thus accustomed to habitual familiarity with intervals which are never strictly in tune. No one can doubt the tendency of such constant association to impair the sensitiveness to minute differences of pitch on which delicacy of musical perception depends. Evil communications are not less corrupting to good ears than to good manners. I feel convinced that we have here the reason why so comparatively few of our trained vocalists, whether amateurs or professionals, are able to sing perfectly in tune. The untutored voice of a child who has never undergone the ear- spoilmg process, often gives more pleasure by the natural purity of its intonation, than the vocalization of an opera- singer who cannot keep in tune. The remedy is to practise without accompaniment, or with that of an instrument like the violin^, which is not tied down to a few fixed sounds. Even with the ^ That a violinist can play incre intervals lias been established by Professor Helmlioltz by the following decisive experiment, performed with the aid of Herr Joachim. A harmonium was employed which had been tuned so as to give pure intervals with certain stops and keys ; and tempered intervals with others. A 208 ESTABLISHED MUSICAL NOTATION. [X. § 113. pianoforte something might be done, by having it, when intended to be used only in assisting vocal practice, put into perfect tune in one single key, and using that key only. The services of such an instrument would, no doubt, be comparatively very restricted, but this might not be without a corresponding advantage, if the vocalist were thereby compelled to rely a little more on his own unaided ear, lay aside his corks, and swim out boldly into the ocean of Sound. 113. The musical notation in ordmary use evi- dently takes for granted a scale consisting of a limited number of fixed sounds. Moreover, it indi- cates, directly, absolute pitch, and, only indirectly, relative pitch. In order to ascertain tlie interval between any two notes on the stave, we must go through a little calculation, involving the clef, the key-signature, and, perhaps, in addition, ' acci- dental' sharps or flats. Now these are complica- tions, which, if necessary for pianoforte music, are perfectly gratuitous in the case of vocal music. The voice wants only to be told on what note to begin, and what intervals to sing afterwards, i.e. it is con- string having been tuned in unison with a common tonic of both systems, it was found that the intervals played by the eminent violinist agreed with those of the natural, not with those of the tempered scale. X. § 113.] ESTABLISHED MUSICAL NOTATION 209 cerned with absolute pitch only at its start, and needs to be troubled with it no further. Hence, to place the ordinary notation before a child who is to be taught to sing, is like presenting hini with a manual for learning to dance, compiled on the theory that human feet can only move in twelve different ways. Not only does the established notation encumber the vocalist with information which he does not want ; it fails to communicate the one special piece of infor- mation which he does want. It is essential to really good music that every note heard should stand in a definite relationship to its tonic or key-note. Now, there is nothing in the established notation to mark clearly and directly what this relation ought, in each case, to be. Unless the vocalist, besides his own 'part,' is provided with that of the accompaniment, and possesses some knowledge of Harmony, he can- not ascertain how the notes set down for him are related to the key-note and to each other. The extreme inconvenience of this must have become painfully evident to any one who has frequently sung concerted music from a single part. A Bass, we will suppose, after leaving off on F i^, is directed to rest thirteen bars, and then come in fortissimo on his high E\ It is impossible for him to keep the absolute pitch of Fi^ in his head during this long interval, which is perhaps occupied by T. 14 210 ESTABLISHED MUSICAL NOTATION. [X. § lU. the other voices in modulating into some remote key ; and his part vouchsafes no indication in what relation the E"^ stands to the notes, or chords, im- mediately preceding it. There remains, then, nothing for him to do but to sing, at a venture, some note at the top of his voice, in the hope that it may prove to be E?^ though with considerable dread, in the opposite event, of the conspicuous ignominy of a fortissimo blunder. The essential requisite for a system of vocal notation, therefore, is that, whenever it sjDecifies any sound, it shall indicate, in a direct and simple manner, the relation in which that sound stands to its tonic for the time being. A method by which this criterion is very completely satisfied shall now be briefly described. 114. The old Italian singing-masters denoted the seven notes of the Major scale, reckoned from the key-note uj)wards, by the syllables do, re, mi, fa, sol, la, si, pronounced, of course, in the continental fashion. As long as a melody moves only in the Major mode, without modulation, it clearly admits of being written do^TL, as far as relations oi pitch only are concerned, by the use of these syllables. The opening phrase of ' Rule Britannia,' for instance, Avould stand thus : do, do, do, re, mi, fa, sol, do, re, re, mi, fa, mi. X, § 114.] TOXIC SOL-FA XOTATIOX. 211 In order to abridge tlie notation, we may indi- cate each syllable by its initial consonant. The ambiguity which would thus arise between sol and si is got rid of by altering the latter syllable into ti. In- order to distinguish a note from those of the same name in the adjacent octaves above and below it, an accent is added, either above or belovr the corresponding initial. Thus cV is an octave above d ; d^ an octave below d. Where a modulation, i. e. a change of tonic, occurs, it is shown in the following manner. A note necessarily stands in a two-fold relation to the out- going and the in-coming tonic. The interval it forms with the new tonic is different from that which it formed with the old one. Each of these intervals can be denoted by a suitable syllable-initial, and the displacement of one of these initials by the other, represents in the aptest manner the super- session of the old by the new tonic. The old initial is written above and to the left of the new one. Thus y mdicates that the note re is to be sung, but its name changed to fa. As this is a somewhat difficult point a few modulations are appended, ex- pressed both in the established notation, and in that now under consideration. The instances selected are, from C to G\ from C to F -, from ^ to C ', from G toi^S. 14—2 212 TONIC SOL-FA NOTATION. X-§1I5. B'^S m ^^^EET= §^t=l??:^'5Ep=: / m d ^Q: lO^^E 8£ mf i^^mm Immediately after a modulation, the ordinaiy syl- lable-initials come into use again, and are employed until a fresli modulation occurs. It will be seen at once, that the difficulty of ' remote keys,' which is so serious in the established notation, thus alto- gether disappears. For instance, a vocal phrase from Spohr's * Last Judgment,' which in the estabhshed notation is as follows, Eeis= i».-^^-f :p-p=:: takes, in the notation before us, the perfectly simple form, s I t \ d m f s \ s f I I I s \f 711. As another exam]3le, take the following, from the same work. --1 f r r 8 t^ d d ^»'/^/ ^ 115. The system of notation, of which a cursory sketch has just been given, originated, it is said, with two Norwich ladies named Glover, but has X. §115.] TOXIC SOL-FA XOTATIOX, 213 received its present form at the Lands of Mr J. Curwen, to whom it also owes the name of '■ Tonic Sol- Fa,' by which it is now so widely known. As it is no part of the plan of the present work to go into technical details, only so much has been said about Mr Curwen's system as was necessary to enable the reader to grasp its essential principle. No mention has been made of the notation for Minor and Chromatic intervals, nor of that for de- noting the relations of time by measures appealing directly to the eye, instead of by mere symbols. On these and all other points connected with his system, Mr Curwen's published works on Tonic Sol- Fa give full and thoroughly lucid and intelligible explana- tions. Mr Curwen has also created a very extensive literature of the best vocal music, printed in his own notation ; given a most remarkable impulse to choral singing ; and estabhshed a system of graded cer- tificates examinations, guaranteeing the attainment, by their holders, of corresiDonding stages of musical cultivation. I have enjoyed some opportunities of watching the progress of beginners taught on the old system, and on that of the Tonic Sol-Fa, and assert, with- out the slightest hesitation, that, as an instrument of vocal training, the new system is enormously, overwhelmingly, superior to the old. In fact, I am 214 TONIC SOL-FA NOTATION. [X. § 115. prepared to maintain tliat the complicated repiilsive- ness of tlie pitch-notation, in the old system, must be held responsible for the humiliating fact that, of the large number of musically well-endowed j)ersons of the opulent classes who have undergone at school an elaborate instrumental and vocal training, com- paratively few are able to play, and still fewer to sing, even the very simplest music at sicjlit. Set an average young lady to accompany a ballad, or to sing a psalm-tune she has never seen before, and we all know what the result is likely to be. Now, there is no more inherent difficulty in teaching a child with a fairly good ear to sing at sight, than there is in making him read ordinary print at sight. A vocalist who can only sing a few elaborately pre- pared songs ought to be regarded as on a level with a school-boy who should be unable to read except out of his own book. If evidence be wanted to make good 'this assertion, it is at once to hand in the fact that the youngest children, when well trained on the Tonic Sol-Fa system, soon obtain a power of steady and accurate sight-singing, and will even tell you whether a new tune pleases them or not, after merely glancing through it, without uttering a note. The reader will please to observe that the above remarks are strictly limited to the achievements of X. § 116.] TONIC SOL-FA NOTATION, 215 the Tonic Sol-Fa system in teaching singing. I express no opinion as to the applicabiUty of its nota- tion to instrumental music, nor do I wish to maintain that even in the vocal branch it has arrived at abso- lute perfection. On the contrary, I am doubtful whether its time-notation, when applied to very com- plicated rhythmic divisions, does not become more difficult than the system in ordinary use, and I con- sider the notation adoj)ted for the Minor mode to be capable of decided improvement. On the main j)oint, however, viz. the decisive superiority of its pitch- notation over that of the established system, and the vitally important consequences as to purity of into- nation which necessarily follow from this superiority, I desire to express the most confident and uncompro- mising opinion. IIG. In closing the enquiry which occupies the preceding chapters, it will be advisable to examine, very concisely, the bearing of our principal result, the theory of consonance and dissonance, on the aesthetics of music. Dissonance was shown to arise from i^pid beats, and the concords were classed in order, according to their more or less complete free- dom from dissonance ; the octave coming first, fol- lowed by the Fifth, Fourth, Major Third and Sixth, and Minor Third and Sixth. This classification was strictly p/?^szca?, depending exclusively on smooth- 216 ■ SMOOTHXESS OF EFFECT. [X. § 117. ness of combined effect. On its own ground, there- fore, it is absolutely unassailable, and whoever says, for instance, that a Major Third is a smoother con- cord than a Fifth or octave, asserts what is as demonstrably false as that the moon goes round the earth in an exact circle. Nevertheless, it by no means necessarily follows that the smoothest con- cords must be the most gratifying to the ear. There may be some other property of an interval which gives us greater satisfaction than mere consonance. Assuming, for the moment, that such a property does in fact exist, the ear, if called on to arrange the consonant intervals in the order of their pleasantness, might very well bring out a different arrangement from that adopted by physical science on grounds of smoothness alone. -Esthetic considerations come in here, with the same right to be heard as mechanical considerations have within their own domain. 117. Now unquestionably the ear's order of merit is not the same as the mechanical order. It ^^laces Thirds and Sixths first, then the Fourth and Fifth, and the octave last of all. The constant ajDpearance of Thirds and Sixths in two-part music, compared with the infrequent employment of the remaining concords, leaves no doubt on this point. In fact these intervals have a peculiar richness and perma- nent charm about them, not possessed by the Fourth X. § 117.] SMOOTHNESS OF EFFECT. 217 or Fifth to anything like the same extent, and by the octave not at alL The thin effect of the octave nndoubtedly de- pends on the fact that every partial-tone of the higher of two clangs forming that interval, coin- cides exactly with a partial-tone of the lower clang. Thus no new sound is introduced by the higher clang ; the quality of that previously heard is merely modified by the alteration of relative intensity among the constituent partial-tones. Major and Minor Thirds bring in a greater variety of pitch in the resulting mass of sound than does the Fifth ; but this can hardly be said of the Major and Minor Sixths compared with the Fourth. On the wdiole, I am inclined to attribute the predilection of the ear for Thirds and Sixths, over the other concords, to circumstances connected with its perception of hey- relations, though I am not able to give a satisfactory account of them. The ear enjoys, in alternation with consonant chords, dissonances of so harsh a description as to be barely endurable when sustained by themselves. This constitutes a marked distinc- tion between it and the other organs of sense. A stench is not improved by alternating with the most fragrant odours, nor nauseous food rendered palatable when administered at intervals between the most dehcious plats, A kick remains a kick, even though 218 RICHXESS OF EFFECT. [X. § 117. it be preceded and followed by caresses ; and repulsive bideousness forms no welcome element in pictorial or plastic art. As instances of tbe kind of discords in wbicli tbe ear can find deligbt, take tbe following. Tbe cbord marked * sbould in eacb case be played first hy itself y and tben in tbe place assigned to it by tbe composer. Tbe effect of tbe isolated discord is so intensely barsb, tbat it is at first difficult to under- stand bow any preceding and succeeding concords can make it at all tolerable ; yet tbe sequence, in botb tbe pbrases cited, is of tbe rarest beauty. \ \ - j^,^ Last Chorus, BacWs Po- J 1^ ^ I ,""^ *^ Passion" (St. Mattheic), — ^ — y — 1/ — ,^>— ^— f>— ^ f^ — •- > ^ \ \ V .. ^ ' --- ± A ^j . -.> ,^ ^ ' I —^-\y 4.-- p ^ /=T -^ ^ g-f=^ Considerations sucb as tbose just alleged tend to sbow tbat, wbile pbysical science is absolutely au- tboritative in all tbat relates to tbe constitution of musical sounds, and tbe smootbness of tbeir combi- X. § 117.] IMPORTANCE OF DISCORDS. 210 nations, the composer's direct perception of wliat is musically beautiful must mainly direct him in the employment of his materials. It would be a serious error to force upon him a number of rules planned, on scientific principles, to secure the maximum smooth- ness of effect; since mere smoothness is often a mat- ter of extremely secondary importance, compared with grandeur of harmony, and masterly movement of parts. The nature of the subject may sometimes call for a mode of treatment needing exceptional smoothness. In such a case the rules may become of considerable importance. It is well, therefore, that a composer should know and be able to handle them, but he should never allow them to fetter his freedom in wielding the higher and more spiritual weapons of his warfare. GAMBIUDGE : PKINTED BY C. J. CLAY, II.A. AT THE UNIYEBSITY PRESS. 1 >; T^ 30 DAY USE RFTURN TO DESK FROM WHICH BORRo . MUSIC LIBRARY This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. ^'m 4 1976 LD21A-10m-10,'74(S1945L) General Libranr Umrenity of California Berkeley f/ ,;>*<<%. » ^m^ iV W l^?^ m- ^^^^