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LESSONS IN CHEMISTRY.-No. XXVI. into the funnel-shaped tube, and thence to the lower end of (Continued from page 381, Vol. IV.)

the refrigerator. He will observe that water emerges from

the other lateral bent tube and falls into a receiver. He will ALTHOUGH refined methods of producing condensation of finally recognise the conditions and general arrangement of vaporised products will be mentioned presently, I must first the apparatus to be such that a continuous stream of cold state that a very large number of chemical distillatory opera. water is made to arrive in contact with the central glass tube, tions may be conducted by driving the vapour to be condensed where, becoming heated, it rises to the upper portion of the into a flask of receiver, and cooling the latter by means of a metal tube, runs through the bent siphon-like pipe, and falls piece of filter paper, wetted by the dropping upon it of water into a receiver. from a receptacle placed above, as represented in the accom- This instrument is of great utility to the chemist. It pos. panying diagram, fig. 20.

sesses very many advantages over the worm and tub. In the
Fig. 20.

first place, it admits of being made of glass, whereas the tub-
worm is almost necessarily made of metal. "Occasionally I am
aware it is made of earthenware, as, for example, when em-
ployed in the distillation of nitric and hydrochloric acids, either
of which would act upon metal. In the second place, the
central tube in Liebig's apparatus admits of being readily
cleansed by friction, and the eye glancing along a straight
orifice can satisfy itself as to the state of purity; whilst, on
the contrary, the cleansing of a tub-worm can only be effected
imperfectly by means of rinsing. Thirdly, the apparatus of

Liebig furnishes a continuous stream of cold water, whereas
'the tub-worm is merely cooled by contact with a stationary
mass of water continually growing hotter. Finally, it is
cheap, whereas the tub-worm is expensive; so regard it from
what point of view we will, the advantages are all in favour of
the apparatus of Liebig.

Occasionally, however, water, although applied under the
conditions most favourable to the exercise of its cooling pro-
perties, is not cold enough, in which case ice must be used, or
a mixture of snow and ice, or, finally, some of those artificial

cooling mixtures known to chemists. Many substances can Whenever the extemporaneous manner of procedure will only exist whilst exposed to excessive cold. Absolute pure not succeed, more elaborate methods of effecting condensation prussic acid must be condensed by a mixture of ice and salt, must be had recourse to. Amongst these the most elegant, and preserved surrounded by ice.' Remove the icy protector, the most advantageous, and in every respect the best for all and the substance escapes in vapour, cases in which the cooling power of mere water suffices, is the These general remarks will suffice to demonstrate the fact refrigeratory apparatus fuist devised by Baron Liebig, and that the degree of cold necessary to produce condensation known by his name. It is represented in our diagram, fig. 21, altogether depends on the volubility of the substance under attached to the beak of a retort heated by a gas flame; in other treatment. Generally speaking, the cooling agency of water words, the whole apparatus is represented in action.

well applied suffices; occasionally the greater cold of ice is

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The refrigerator consists of a metal tube, tin plate, brass, required, and, still more rarely, artificial cooling mixtures. or copper, about 24 inches in diameter, and not less than two During some of his experiments on the condensation of gases, feet long, through which passes a glass tube a little longer Professor Faraday applied cold of such intensity that it is than itself, and retained in the middle of the metal tube by represented by some 240 or 250 minus degrees of Fahrenheit, means of two perforated corks, one at either end, the punctures whereas the freezing point of water is + 32o on the same being rendered water-tight by means of white lead cement. scale! In addition to the modifications of the distillatory

If the student now glances his eye at the metal tube he will process already detailed, there are others dependent upon the discover that two bent tubes are aitached to it externally and nature of peculiar substances operated upon. Oil of vitriol, laterally, one towards either extremity. The student will for example, is exceedingly difficult to be distilled in glass also observe that water from the barrel or reservoir B trickles vessels. Its boiling point is high and its vapour is evolved in

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explosive starts dangerous to the safety of the apparatus ; yet | tion of which receives the sublimed sokid, whilst the accom-
oil of vitriol may be safely distilled from a glass retort into a panying liquid sinks into the flask underneath.
glass receiver by heating it in contact with fragments of These general descriptions will suffice to give the reader a
platinum foil, and applying the distillatory heat in such a general notion of the nature and objects of distillation. The
manner that the surface rather than the inferior strata of the process, he will see, is one of remarkable simplicity, yet, strange
Auid shall be vaporised,

enough to say, it appears to have been unknown to the
And first as regards the platinum foil, let me illustrate the ancient Greek and Romans, its origin being due to the
effect of this substance comparatively. The reader has pro- Arabs.
bably observed a glass of champagne which had ceased to In stating that the Romans were unacquainted with distil-
effervesce, restored to its condition of primary effervescence by lation, I must be understood to limit the expression to distilla-
dropping into it a crumb of bread, a bit of cork, or almost any tion in our sense of the term, understanding it to apply to the
angular body. Now it is a general function of all solids, more compound operation of raising a liquid in vapour, and after-
especially pointed and angular solids, to promote the effer- wards condensing that vapour in a refrigeratory apparatus.
vescence and ebullition of liquids into which they may be Nevertheless, a rude sort of distillation was occasionally fol.
immersed ; and, in this respect, metals, beyond all other lowed by holding wool over the ascending fumes of a vapo-
bodies, are pre-eminent. Platinum, however, is one of the rising body, and thus effecting their partial imbibition.
very few metals on which oil of vitriol does not act, and for
this reason it is employed. The rationale of the action of
solids in promoting the ebullition of liquids is not yet fully
understood, although the result is well known and taken ON PHYSICS OR NATURAL PHILOSOPHY.
advantage of in many important operations.
The peculiar method of applying heat so as to exercise its

No. XXVII.
force on the upper rather than on the under layers of a liquid
is represented by the annexed diagram, fig. 22, which repre-

(Continued from page 379, Vo!. IV.)

THEORY OF MUSIC.
Fig. 22,

Language of Music.-In arranging musical sounds in regular
order, it has been already observed that, after having ex-
pressed a certain number of them according to an ascending
or descending scale, all other sounds are only a reproduction
of these, modified only by their passing from low to high, or
from high to low; so that they are still recognised to be the
same sounds.

The smallest distance which separates two sounds of this kind is called an octave. In the interval from one octave to another, it is easy to distinguish twelve different sounds placed at equal distances from each other. This distance is called a semitone, and the series of these twelve semitones is called the Chromatic Scale, from the coloured mark used by the Greeks to point out that the music was to be softened by lowering the sounds half a tone.

This series, sents a retort supported by a hollow truncated cone of sheet- however, was simplified by the reduction of the scale to the iron let into a charcoal furnace in such a manner as to fit it seven natural notes or sounds of the Diatonic Scale

, so called almost exactly. This arrangement is such that glowing char- because they were produced by the transversal vibrations of coal being placed between the cone and the furnace, as repre. strings stretched across any hollow musical instrument. To sented in the diagram, will heat the

surface layers of the oil these seven sounds was added an eighth, which was the reof vitriol almost exclusively. As regards the refrigeratory petition of the first sound, in order to complete the octave. In part of this apparatus, the reader will see that a simple flask this scale, instead of proceeding regularly by equal semitones, without any contrivance for wetting it is alone employed. a series of full sounds and of alternate semitones was adopted In point of fact, the condensation of oil of vitriol does not and its name, Gamut; is evidently derived from the French require such artificial aid, which, instead of proving service- word gamme (which means the same thing), with the addition able , would, in all probability, crack the receiver.

of ut, the name of the first or lowest note in the natural Occasionally a volatile solid is the subject of distillation, in

scale. which case the term sublimation is applied. The reader has

The sounds of the Gamut were originally indicated by more than once, whilst engaged in the investigation of arsenic, letters, of which the lowest was the Greek T, Gamma, cor. gone through the process of sublimation. Sometimes the responding to our C; hence, the French word gamme, as the result of distillation is partly a solid, partly a fluid, in which name of the scale of music. In the eleventh century, the diagram, fig. 23, is occasionally employed, the globular por- ut, , mi, fa, sol, la ; and these names were taken from the

first syllables of a religious hymn. Five centuries later, the
Fig. 23.

note named si was added, and then ut repeated, which com-
pleted the series and the octave, as follows:-
Letters

C, D, E, F, G, A, B, C,
Names

ut, , mi, fa, sol, la, si, ut.
Notation. There are two kinds of signs employed in musical
notation ; viz., those which express intonation, and those which
indicate time or duration. The former are the notes ; each note
represents a distinct sound, and its value, that is, the time
during which this sound is emitted. Five parallel straight
lines at equal distances, with marks for the notes placed alter-
nately on the lines and the spaces between them, constitute
what is called the staff or stave, as in fig. 140, where the first
octave is represented with the treble clef; the first note C
being put upon a portion of a line called a ledger line. To
this note, which is called ut, English musicians give the name
of do.
The place

wkich a note occupies on the staff determines its
intonation. The lower it is placed, the lower is its sound; and the
higher it is placed, the higher is its sound. By a note being

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placed higher or lower here is meant, placed nearer to or further divided into tuo Quavers ; the Quaver into two Semiquavers; and
from the bottom of the page of a book, or of the face of a the Semiquaver into tro Demisemiquavers; and so on. The
board placed vertically, on which the lines of the staff are general rule for the relation subsisting between the length of
drawn horizontally. It has been observed in our last lesson, these notes is, that each of the longer is double that of the
that the terms low and high, with respect to sounds, are arbi- shorter which follows it. The following table, fig. 141, ex.
trary and conventional; and that the real difference by which hibits the equivalent values and relative lengths of each note,
they are distinguished consists in this, that the sonorous body with their corresponding times, and parts of a time.
which yields a low sound makes, in a given time, a less num- The Breve is a note seldom used, but of course it would con-
ber of vibrations than that which emits a high sound. The tain eight times; it was denoted by the same character as a semi-
notes placed on the lower part of the staff represent, therefore, breve with two bars on each side of it, as shown in p. 183, vol.iv.

col. 1, line 19. The measure containing four times may be con-
Fig. 140.

ceived as divisible into two equal parts; but there are also
pieces written in the measure containing two times. For this
purpose no change is made in the system of signs just
explained, except that the full measure, instead of being
represented by a semibreve, is represented by a minim

or its equivalents. As to the measure of three times,
Letters . C, D, E, F, G, A, B, C,

it is represented by a minim and a point alongside of it, as Names · · do, re, mi, fa, sol, la, si, do.

shown in the page and column above referred to, line 26, the

point being employed to denote the prolongation of the dura.
sounds which are relatively lower than those do which are tion during a third time. The point is used to lengthen all
placed above them. The system of the gamut thus indicates the other values of the notes ; so that, when placed alongside
that the notes rise to a greater height, as they are placed suc- a crotchet, a quaver, etc., it adds to their duration one-half of
cessively and alternately ascending from the lowest or first its original value.
line of the staff;.then, on the space between the first line and The Gamut.-We have said that the diatonic scale proceeds
the second ; then, on the second line; next, on the second by tones and semitones. By a full tone is understood the
space; again, on the third line; next, on the third space; and distance which separates any degree from that which imme.
80 on. Yet, the distance which separates the sounds thus diately follows it, when this distance can be easily divided by
expressed is not regular ; sometimes it represents a full sound, the ear into two distinct sounds. In the contrary case, the
sometimes a semitone, as we have said, when speaking of the interval is only a semitone. The diatonic scale is divided into

Sometimes the intervals between these successire five full tones and two semitones. In order to understand the
sounds are called degrees. Two notes placed on the same line position of both, we must consider the octare as composed of
are said to be in unison; the interval from one degree to that iwo equal parts, containing each four notes or degrees. Each
which immediately follows it is called a second; the interval of these parts, considered by itself, comprises two full tones
from the first degree to the third is called a third; and so on and one semitone, to which must be added a full tone which
to the interval from the first degree to the eighth, which is separates the one from the other. Thus, in the first part, which
called an octave ; as explained in our last lesson. So much for extends from do or ut to fa, we find from uut to ré' a full tone, from
intonation.

ré' to mi a full tone, and from mi to fa a semitone. In the A measure, in music, is a space of time, at the end of which second, from sol to ut, we find between sol and la a full tone, the ear feels the want of rest; this rest forms the starting point from la to si a full tone, and from si to ut a semitone ; lastly, it of the following measure. The smallest portion of melody is we join the two series, we find between fa and sol a full tone; always divisible by the ear into a certain number of measures, which completes the construction of the octave. Whether this and each measure is divided into three or four times. The series of sounds be a consequence of the organisation of our audio times are also divided into parts which may be either irregular tory apparatus, or be the result of convention or of custom, it is or symmetrical, and this is what constirules rhythm. In the no less true that the notes thus regulated are generally adopted

gamut,

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system of notation, every measure is indicated by a bar per- ' and satisfy all ears. On this point, therefore, siz, the construc-
pendicular to the straight lines of the staff; consequently the tion of the scale, all the combinations and thrilling effects of
assemblage of ralues comprised between two bars of this kind modern music depend.
ought always to represent one equal period or duration divisible Our remarks on the structure of the scale are especially appii.
into equal times. The longest measure is that containing four cable to that which begins at the nose do or ut, and which is con-
times. The note which represents it in its whole duration is sidered as the normal scale. Bat if we begin with any other
called a Semibrere. If this measure be divided into two notes note, with re' for example, it is plain that, in order to preserve
of equal length or duration, these are call Minims; conse- the same ratios between the tones and semitones, it must be
quently, two Minions are equal to one Semibrere. When the made to undergo a modiácation according to the intervals
same measure that of the semibreve) is divided into four established in the scale of ut. These ratios are such that the
notes, each of them is called a (rotchet. The Crotchet is semitones are found besteen mi and fa, and between si and a

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Lhat is, between the third and fourth degree, and between the third is minor, that is, composed only of a tone and a semiton, seventh and eighth dezree.

as may be seen in the following example : fig. 142. Now, in the scale commencing with re' we should have a full

Fig. 112. tone between re and mi, but between mi and for only a semitone. It will be necessary to raise fa by a semicone, and between

Mode mejor. Mode minor. Made major. Mode minor. this elevated fa and sol, that is, from the third to the fourth liegree, the semitone is necessarily found re-established in its proper position. The same thing will take place between si and #l, which are separated by a semitone in the normal scale; but in raising ut by a semitone, the order from the sixth to the seventh degree, and from the seventh to the eighth degree, would

These two modes are cornected with each other by very be equally re-established. These modifications are effected by natural reciprocal relations, which exist either between the the aid of two signs, the sharp, # and the flat, b. The sharp tones which bear on the clet the same accidents, or between the raises By a semitone, the note Uefore which it is placed; the scale major and the scale minor of the same fundamental note. fia', on ihe contrary, lowers it by a semitone. When the note Thus the tone of ut major has fur its relative minor that of la sharpened or flatlened is required to return to its normal or

minor, because neither bears any accident on the elef. The natural intonation, we prefix to it another sign, called the tone of re' major (with two sharps) has for its relative si minor; natural, viz. 4. In this manner, the necessary order oLiones the tone of re' mirror (with a flat), is the relative of the lone of and semitones in any scale can be fixed, and consequently as a major, ete. Every scale can pass from the mode major to many scales as there are'ones in the whole extent of the ociave the mode minor, and reciprocallý, without changing the fun

damental note. can be obtained. In order not to unnecessarily load the notation

For this purpose, it is sufficient to inodify tha with signs, we place ai the beginning of any piece of music, signs of the clef so as to re-establish the required intervais hy the accidents (that is, the sharps, the flats and the naturais), and the flats as negative signs, we pass from the mode minor to

either mode. The sharps being considered as positive signs which ought to affect the modified noies throughout, and we give to each scale thus fixed the name of the note with which the mode major, by adding to the clef three positive signs ; it commences. For example, when we say that a piece of and in order to convert the mode major into that of minor, we music is in re'

, we mean that re' is the first low note, add to the clef three negative signs. Moreover, it is evident or the fundamental note of the scale in which it is that effacing the fats is equivalent to adding the sharps, written.

and effacing the sharps is equivalent to adding the fiais.

Examples : fig. 113. Modes. -Our preceding observations rela:ing to the structure

Fig. 14 of the scale have reference to the scale of it as the model ; we shall add here, of ut major, and we employ this phrase of

To pass frou sol major To pa s from ré minor

'inin ré majar. transition in order to explain what is meant by mode. The word mode is applied io two different arrangements of the we scaic--modifications which are very slight at bottorn, but which comuuuicate to it a churacter and properties entirely different. The language of music possesses two modes: the majir anå t)e ininor. The former is especially applied to the expresbion of joy, happiness and expanded sentiments; the opher From this arists in inportant point in the reading of inusia, expresses sadness, grief, and dark, close, and contracted sculi. namely, the determination of the tone of the fundamental mute thents. These two modes, which are so different in their of the scale in which the piece is written. By the inere (ffects, differ from each other only by a slight alteration in the inspection of the signs placed on the clef, every musician must ystem of the scale. We have said that in the m de majir, know what is the scale which is used as the foundatior of the the senitones are situated between the third and the fourth piece which he is about to perform. These signs are suljece and between the seventh and the eighth degrees; in the mode in their arrangement to rules so symmetrical that they are minor, they are situatid between the second and third, and easily fixed in the memory as soon as they are studied with berween the seventh and eighth. Cunsequently, the difference attention. The following is a summary of the preceding cossists simply in this, that in the mode ininor, the first semi- details in a few words :lone is situated between the second and third degree, instead The first charp placed on a clef is always a fa. The followof between the third a:d fourth.

ing sharps are placed from fifth to fifth in the ascending scale: The normal scale of the mode ininor is that of la. In com

dout, sol, rt, la, mi, si, mercing with this note, we find a semitone between si and ut,

In the sharpened tones the fundamental note or tonic major that is between the second and third degree. In following the is always a semitone below the last sharp placed on the

clef. ascending scale, we meet a semitone between mi and fa, and a full tone between sol and la final. This has been the source of

In the fiattened tones the first flat is placed on the si. The many disputes among musicians. Some have raised the fa as

following flats are placed from fifth to tifus on the descending hotel as the sol by means of a sharp ; others have preserved the scale ; si

, mi, la, re, sol, ut, fa.

In the flattened tones the fundamental note is always five unade no alteration, so that the semicne remained in its natural degrees below the last flat; if there be several Hats, the fundas place between the sixth and seventh degree. Thus, there are mental note rests on the last but one. three ways of terminating the ascending scale minor, and three

The fundamental nule of a relative minor is always two effects ; but it is to be especially observed that the principal same fundamental note minor, by adding to the clef three enges of employing them at pleasure, but with very different degrees above the fundamental note major, and inversely.

Transition is made from a fundamental note major to the character of this scale consists in the interval from the second to the third degree, which is always a semitone. All this negative signs, cr their equivalent.

Transition is made from a fundamental note minor to the proves, however, that the termination of the ascending scale minor is not an essential characteristic ; but in descending this same fundamental note major, by adding to the clef thrce scale we can alter it as well as in ascending ; and what is more

positive signs, or their equivalent. remarkable is, that the interval between the seventh and eighth

One point often embarrasses the student, respecting the degree

, which is called the sensible note, and which is almost forced determination of the tone ; and this is the choice between the to be a semitone in ascending, is more grateful to the ear in the node major or the mode minor represented by the same signs descending scale when it is a fuil tone.

on the clei. One simple method of determining it is this:

when the fundamental note is not clearly fixed by the first The fundamental difference between the inoide major and or the last concords of the piece, we search for what would be the mode minor is expressed by saying that, in the foriner the the sensible note (the oth degree) of the tune if it be minor. In third (the interval between the first and third degree) is major, this case, the sensible noie would be necessarily affected with that is, composed of two full tones ; and that in the latter, the an accident (positive or rega:ire) in the course of the first

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concord. If it be not so, the piece is evidently in the mode Maëlzel has generally been adopted. This instrument is major.

composed of a pendulum which is put in :aotion by a clock, Rests.-Returning to the subject of the signs and measure and which divides a unit of time, the mir:ute, for instance, of duration, we find that the signs of duration are of two kinds ; into a greater or less number of parts, according to the length the one applied to the expression of the notes, and the other given to the rod. This length has a certain ratio to the to that of the rests or pauses. The former have been explained, duration which the division of the measure taken as unity the latter have some analogy to them. The following table, must represent, and it is regulated by means of a bob whieh is fig. 144, shows their form and their relations :

made to rise or fall on the rod, which is graduated accordFig. 114.

ingly. Bemibreve. Minim. Crotchet, Quaver. Scini.

Signs of Expression—The signs of musical expression have quaver. semiquaser. for their object the indication of those varied shades of feel.

ings, such as the sweetness, force, etc., of the notes, which give to music, so to speak, its physiognomy, and enable it to

represent all the different emotions of the soul. The following 7 포

signs are employed in the regulation of such expressions ;

some are applied to the intensity of the sounds; others to the Semibreve Minim Crotchet Quaver quaver smiquaver

association of the notes; others to the modification of the

general movement. The first set vary between the limits of Points may be added to the rest-signs, and they serve also pianissimo (the softest or sweetest), expressed by PP., and to increase them by half their value. It is plain, as this table fortissimo (the strongest or loudest), expressed by FF. A single shows, that the values of the notes and the values of the rests Pindicates piano (soft or sweet), as forte, denoted by a single F, are the same in point of duration, and are employed to com

indicates strong or loud. P and f themselves are also modiplete the measure in the same proportions. Their difference fied by the signs sf or rf (that is, &forzando or rinforzando), or consists in their expression being negative instead of being by dim. (diminuendo), cres. (crescendo), cal. (calendo), etc. "The positive.

signs < or > express the increase or decrease of the force of a Signs of Measure.—The measure of a piece of music is indi- single note or of a member of a passage in music. The lozenge cated on the clef by the following signs. The measure of two

<> indicates a crescendo of small extent, or a note merely times, that is the binary measure, is expressed by a C, or by strengthened. Two signs are sufficient to denote whether the the figures i ; that of four times, by a C crossed vertically by notes are connected or detached. The curve ~ indicates that a bar, thus ¢ ; the measure of three times, that is the ternary the notes which they unite must be run together or connected measure, is expressed by, or only 3, which signifies that it in their utterance, and the points placed above each note is composed of the value of three crotchets. An important signities that these notes must be detached. When the points variety of the measure of two times, or binary, is that which is thus separated are surmounted by a curve ..., the notes marked %, and of which each time is divided into three qua- besides being connected must be articulated in a sensible vers. In the measure of four times, or quarternary, marked 1,2, manner. Lastly, the momentary acceleration of the general each time is divided into three quavers, which is called a movement is indicated by the letters accel. (accelerando); or triplet. The measure : is one of three times, each being repre- they are moderated by these, ritard. (ritardando). When a sented by a quaver. General rule :-In every measure ex- rest greater than its normal value is to be made on a note, it pressed by two figures of which one is placed above the other, is surmounted with the sign I which is called a point of the upper figure indicates how many times this measure must contain the division of the measure of four times expressed by Clefs.-The clefs or keys are chiefly required in order to the lower figure considered as unity. Thus indicates that supply the want of extent in the staff. 'Every voice and every the measure contains three times the fourth part of the instrument has its diapason, that is, its own proper extent or measure of four times, or three crotchets ; | indicates that it compass of sound. The limits of each must be marked accordcontains nine times the eighth part of the saine measure, that ingly. The clef is employed to show the extent of the staff, to is, nine quavers; and so on. We must now determine the determine its position in the long series of sounds appreciable proper duration of this division of the measure taken as unity. | by the ear, and to represent in short compass the range There are two methods, the one approximate the other very of the pianoforte. By help of this artifice, a piece writexact. The former, which is the most common, consists in ten for å given voice or instrument can be performed by a placing at the commencement of a piece of music a word different voice or an instrument of a differert diapason. The generally taken from the Italian language, which indicates whole of a piece may be thus transposed into the inedium staff the movement of the composition. The words of this kind, which better suits every instrument or every voice. although many, can be reduced to some standards which are There are three principal clefs : the clef of sol, which is that modified by augmentative or diminutive expressions. Thus, a of the tenor voice, of violins, flutes, clarinettes, hautboys, horns, very slow movement is indicated by the words lento, largo, and the right hand of pianists; the clef of fa, which is com. larghetto; when it is somewhat less slow, it is indicated by the monly used for the bass; it determines the place of fa on the terms grave, adagio, cantabile, moderato ; a more firm and ani. fourth line; when applied to the voice, its position may be mated movement is indicated by the words andante and varied. As to the clef of ut or do, it gives its name to the note andantino; a gay and lively movement, by the terms allegro on which it is placed. The alto of the clef of ut is written on and allegretto; lastly, a quick and rapid movement, by ihe the third line ; its place can also be changed when applied to words presto, prestissimo, vivace, etc. All these words, of which the voice. These three clefs seem placed at the ascending the value is well known to musicians, have been long em- fifth from each other ; yet the medium ut of the clef of fa is ployed to express the movement of a musical piece. Never- at the lower octave of that of the clef of ut, and at the double theless, it is evident that in order to express these different octave of the medium ut of the clef of sol. The following rates of movement exactly, mechanical means must be em table shows the form, the relative position of the clefs in use, ployed. After a great number of trials, the metronome of M. I and the place of the medium ut on their respective staves.

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