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If we compare our systems of notation and tones, 10 far as we have hitherto considered them, with the ow of keys of a pianoforte, or with our representaion at page 6, we shall perceive that we have not as vet learned to know and to write all the tones, and, therefore, that we are not in possession of the whole system. We know nothing of the tones given by the black keys, which are distinguished in the figure above quoted by the numbers 2, 4, 7, 9, and 11. We have allowed ourselves this postponement, in order to secure a good foundation, and shall now proceed to the elucidation of the omitted tones, in doing which we crave reference to our said representation of the key-board at page 6. RAISING.


If we place before any note this sign,

called a sharp, or sign of raising, the key and tone originally indicated by that note, are no longer meant, but the key and tone next above it, whether this latter be a black or a white key. If, for example, we place a before the note c


it is not the key c (fig. page 6, No. 1), but the next higher a black key, No. 2-which is intended. If a sharp were before d, the black key 4 would be understood. If it were before e, the key 6, which we see is a white key, would be meant.

The tone so raised must now also change its name, which is done by simply adding the word sharp after it. Thus c so raised, is called c`sharp, and so of the rest.

There would seem to be 14 of these tones in the octave

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being meant, the note is understood to indicate the key and tone next lower, or deeper, whether that be a black or a white key. If, for example, a b be placed before c, the key c (in the fig. No. 13) is not meant, but the next deeper, No. 12, and, as we see, a white key. If a b be before b, the key No. 12 is not understood, but the next lower, No. 11, which is a black key. Such a lowered tone changes its name in a similar manner as the raised tones, by adding the word flat to the name of the tone. Here we see the tones of an octave,


c flat, b, b flat, a, a flat, g, g flat,

f, f flat, e, e flat, d, d flat, c

Such tones as differ in name only, but are in reality the same in pitch, we call enharmonic. Thus b and cb, e and fb, e and f, b and c, cand db, ab and g, bb and a, and so forth, are enharmonic tones.

It may seem surprising that we should have double names for the tones. Why don't we name the black keys c, d, &c., only, or db, eb, &c., only? Why should e be sometimes fb, and f sometimes e? These apparently superfluous double names, have been adopted for very sufficient reasons. They are quite indispensable to clearness and facility of notation. This will in part be shewn in this work; but it can be done conclusively only in the treatise on composition and the science of music.

The scale in which all the raised and depressed tones are introduced, which we have not already described above, under the name of enharmonic, we call the

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We will not refrain from a remark in this place; simple as it may appear, it will be found of use hereafter. It is this: that the effect of the natural is two-fold, viz., if it remove a sharp, it depresses the tone: if it remove a flat, it sharpens the tone. (D.) DOUBLE RAISING AND DOUBLE DEPRESSION. There are circumstances, which will be explained hereafter, in which it is necessary to raise or depress a tone doubly; so that instead of its own key, the second key from it, either above or below, must be used.

The double raising is signified by a double sharp, which is thus formed :


If a double sharp is placed, for example, before c, neither c nor c is meant, but the key next above that of c (which we have hitherto called d, and is numbered 3 in the Fig. page 6). The name of a tone thus circumstanced is now formed by adding "double sharp" after it; as c double sharp, d double sharp &c.

The double depression is signified thus,-bb If this sign be placed before a note, it will mean the second next lower key. If it were, for example, before d, neither d nor db is to be struck, but c No. 1, in the above Fig-p. 6 The name is now formed by adding "double flat" to the name of the tone so qualified.

Here is an example of double raising and double depressing.

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g, g sharp, g double g, d, d flat, d double flat, d. sharp,

But how, if we want to remove half only of the raising or depressing, and so reduce them to one sharp or one flat? Then we might employ one natural only, which in strictness ought to be sufficient; but in order that no oversight may occur, whereby the natural might be considered as a total removal of the double sign, one of these latter signs is allowed to remain with the natural; thus,—


g, g sharp, g double g sharp, d, d flat, d double d flat,

We perceive, moreover, that by the double sharps
and flats, more names are introduced for one and the
same tone than we have hitherto had (page 14), for
now, every tone may be called by three names; viz.-
c may also be called b sharp or d double flat,
d flat or b double sharp,
e double flat or c double sharp,
which faculty applies equally to all in the same rela-
tive positions. Why these several names are neces-
sary and what object they serve, must be explained

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elsewhere. We will merely take notice of the nam and again impress upon our minds, that tones pos sessed of several names, but of the same pitch in ou system (produced by the same keys), are calle enharmonic tones. Therefore, c, b and dbb, and also c, db, and b &c. are enharmonic tones.

We must confess that the object of the mark natural, cannot be clearly perceived, until we know how long the effect of the sharps and flats generally lasts; but the whole of this instruction on signs

hangs so easily together that it has not been thought

expedient to omit it here.

Now let us return to the


in order to understand them completely.

When we introduced them at page 7, we said that all tones bore their names, or names derived from them. We see now that each degree is susceptible of appearing in five forms, viz., its own original, sharp, double sharp, flat, and double flat; and that it is therefore appropriate for five tones and five names. We therefore will reckon all tones named from one degree as belonging to it. Therefore to the degree belong the tones,


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c, c sharp, c double sharp, c flat, and c double flat, d, d d d d and so forth; although these tones under another name may be reckoned as belonging to another degree; for instance, the tone c sharp, as d flat towards d, and as b double sharp towards the degree b.

Finally, we now only, know all the contents of our tonal system, of which, in the first section, we were acquainted with only the unaltered degrees. We know that every octave, besides the seven degrees (c, d, e, f, g, a, b), contains five raised or depressed intervening tones (c, d, f, g, at, or db, eb, gb, ab, bb), and therefore all together twelve different tones, and we can now under the tonal system understand the contents of all the seven degrees, with their elevations and depressions, through all octaves.


Since in Music, tones must be combined together, their reciprocal relationships must be estimated.

With the most superficial observation, we perceive that one tone is higher than another: that g or a for example, is higher than c, in the same octave. But as there are several tones higher and lower, that difference of height requires to be more precisely determined. The enumeration of the degrees is more exact. We begin at one which we call the first, the next, proceeding upwards, is the second, the following is the third, and so on.

The tones are named in the order of the degrees, first, second, third, &c. up to the octave, which is the same in both successions. Above the octave, however, the tones are called ninth, tenth, eleventh, &c.

It is easy to perceive that the eighth degree is no other than the first, in a higher octave; the ninth nothing but the second in a higher octave, and so forth.

For most persons, it will be sufficient to observe the names of the degrees up to the octave and ninth.

The others do not come into use until late in the study of composition, that is, in double counterpoint. Thus, if C be taken as first, D becomes the second, E the third, F the fourth, and so forth. If F be taken as first, G is the second, A the third, and so forth. Hereby we can give the relationship of the tones more exactly; we can say for example of G, not only that it is higher than C (in the same octave), but also, that it is four degrees higher; that it is the fifth degree, the fifth of C.*

If we compare two tones in respect of their height, we establish them in a relationship towards each other. The general name of that relationship is INTERVAL.

We say, therefore, that C and D form with each other an interval of a second, G and D the interval of a fifth, C with C (figuratively), the interval of a first.

But also, these determinations are not completely sufficient for we already know that each of our degrees includes five different tones. Which of them is now meant? If for instance, we want the fifth of C, do we mean g, or g or gx, or gb, or gbb? They are all on the fifth degree from C, and are therefore all fifths of this tone. Here the enumerations of the degrees and the mere name of the interval, leave us in uncertainty.

We require, therefore, a more exact


and for that end the smallest intervals of our system are taken.

We want two such measures, which we call


A tone is the interval between any immediately adjacent degrees, having in our system some other tone between them. Therefore c and d form the interval of a tone; for they are degrees lying next to each other, and between them there is the tone

or db. In like manner and d, e and ƒ, bb and c are intervals of a tone, since they are on degrees next to each other (c and d on the C and Ď degrees, f on the F degree, bb on the B degree, and therefore in these cases the degrees C and D, E and F, B and C, are next to each other), and between them lies a tone; namely, d between c and ƒ between e and ƒ, and b between bb and c.‡


* According to custom, we count from the lower to the upper. If, on the contrary, we count from above to below, from an upper tone downwards, we put the word inverted before the name of the number. Therefore F is from G the inverted second, from A, the inverted third, and so forth.

+ The doctrine hitherto has been, and it is inculcated also in the former editions of this work, that it is necessary to employ three measures, derived from Acoustics-the tone, the greater semitone, and the lesser semitone. The greater semitone was the interval between two tones, named from adjoining degrees, and without an intervening tone, as for example, between b and c, c and do, and fand g. The lesser semitone, the interval between two tones, named from the same degree, without an intervening tone, as for instance, between 6 and b, c and c, f and fx, and so forth. As a foundation for these three measures, the least perceptible gradation of sound was assumed and called a comma, of which a tone is said to consist of nine, a greater semitone of five, and a lesser of four. But the whole of these distinctions can be dispensed with in practice, and therefore need not be discussed in this work.

Here we perceive the influence of the double names, which we mentioned at pages, 14 and 15; c and d form between them a tone, e and falso; but db and d, or e and gb do not do so: for db and d belong to the same degree; while e and go are not degrees next adjoining to each other, although of the same height as e and f. We must wait till a later period of our instruction, to comprehend why a difference, apparently in name only, should be necessary.

It is therefore apparent, moreover, that in this matter, it is of no consequence whether any of the tones belong to black or to white keys.

A semitone is formed between two tones belonging to the same degree, or belonging to next adjoining degrees, between which, in our system, no tone intervenes, Therefore, b and b, c and c,f and fx, dp and dbb, form semitones; for they belong to the same degree, and there is not in our system any intervening tone between them: and moreqver, no key on our key-board. In like manner, b and c, c and db, f and g, form semitones; for they belong to adjoining degrees, and have no intervening tone.§

With these measures, all the relationships in our tonal system can be given with precision: that is, we can count how many tones and semitones they contain.

If we examine first the tone, for example, c―d, we find that it contains two semitones; viz., c-c, and cd, or, what is the same, c-db and dp—d.

If we examine the third, c-e, we find that it contains two tones: c-d and d-e.

If we examine the seventh, c-b, we find in it 2 Tones, c--d and d—e;

1 Semitone, e-f;

and moreover

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1 Tone,

1 Tone, f

1 Semitone,

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1 Tone, a-b; or 1 Semitone, a We measure therefore as we please, or as it is most convenient.

Now only can we give the relationships of tones with precision. If starting from c we require that seventh which lies five tones and one semitone higher, b alone can have been intended, for bb would have been only four tones (c-d, de, f-g, g-a), and two semitones (e-f, a-bb¶) distant; and therefore a semitone too near, while b would have been six tones distant, viz., c-d, de, e-f

-b, and therefore a semitone too distant. Bbb would be still nearer than bb, and b× still further than b

But it would be tediously minute were we to introduce the measure, whenever we have to speak of the quantity of an interval; therefore we adopt FOUR CLASSES OF INTERVALS,

and can now with one single adjective for each, give its exact quantity. Every interval may be

§ Here also, the enharmonic equality (gleichheit) of b—b, c-c, &c., to b-c, c-db, &c., strikes the eye.

Not every relationship of tone in general can be given. For the mathematics teach us, that all quantity and consequently the interval or distance between two tones is endlessly divisible, and therefore numberberless relationships of tones are possible. Further, it is not the fact that always and everywhere the relationships of tones admitted and familiar with us, have been and are alone used. On the contrary, many quite different relationships have been employed or attempted: as we learn from the history of music. (See the author's article on Greek music. in the Universal Lexicon of Music.)

Cannot we say five tones, or reckon one tone for two half tones? We should then be obliged to reckon so, viz., c-d, d-e, e-ƒ, f—9—, g-a; but a is not the seventh from c, although enharmonically equal to it; and g-bb is not a tone, because the (degrees) g and b do not lie next to each other.

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There does not reign among musicians and authors the desirable unanimity in the naming of these intervals. The major and minor third are also called hard and soft, greater and lesser: the major fifth is also called the perfect fifth, as if any interval could be anything else but perfect. The minor fifth is even called the false fifth, as if it in its proper place, were not, by nature, as true and right as any other interval.

The most remarkable is the naming of the fourth (of two tones and a semitone) c-f as a minor fourth, and the fourth f-b or c-f (of three tones) as a major fourth. These appellations, however, have not only the authority of earlier writers, but also the support of parallel cases: for the major second and third give inverted the minor seventh and sixth, whence it seemed to follow, that the inversion of the major fifth should be called a minor fourth. But these inverted intervals have no practical importance, and it seems more appropriate to consider all the relations of the original degrees in like manner towards or with the first, and all, as major. This gives us the advantage of the above easy method of estimating the intervals.

+ Let us remember that the distance only of the higher tone from the first (c) gives the true intervals. The names of the degrees among themselves produce all kinds of intervals: for example, d-f and e-g

If we should forget any of these quantities, we ha only to extract the interval from the names of t degrees, and measure it.

If we desire to find out a major interval fro any given tone, we first fix upon the proper d gree, then measure the distance, and add or dedu if it be too great or too little. If for exampl we want to know which tone is a major fourt from f; we see, first of all, that the fourth degre from fis b; but f-b contains three tones, an our normal fourth c-f has only two tones an a semitone; consequently we must lessen fb by changing b into bb, and we have then the tone f-g, g-a, and the semitone a-bb. Or if from b we sought a major fifth, we must in the firs place fix upon the fifth degree f, and then measure This latter contains against the normal fifth c-g. three tones (c-d, d-e, f—g), and a semitone (ef), but our fifth contains only two tones (c-d, d-e), and two semitones (b-c and e-f), and therefore is too small. We therefore raise ftof, convert thereby the semitone e-f into a tone, and have acquired our major fifth b-.

When we have once determined a major interval, it is easy to make therefrom, a minor, diminished, are not major thirds; neither is f-b a major fourth, as the above measures will shew.

We think proper to notice here a particular distinction of intervals, which is most ancient in the theory of music; and which, though unimportant as to our main object, ought not to be unknown to the musician or to the lover of art.

In order to understand it, we must remember that in acoustics (page 7) a sound is called higher, the greater the number is of vibrations made by the sounding body in a given time. That if a given tone requires one vibration, its higher octave will require two vibrations in the same time (consequently of double velocity); the higher fifth three vibrations, the second higher octave four vibrations, the next major third over this latter octave five vibrations, the next minor third over the major third, six vibrations. It being granted that a sounding body making one vibration in a second produced the great C, a sounding body which made two vibrations in a second would produce the small c. The sounds would stand in this relation

as C: c : 9: c : e: 9

so 1 : 2 : 3 : 4 : 5 : 6

The next relation would be 6: 7. It would produce a sound which we should call bb, the minor seventh from c.

Now we may understand the distinctions in the ancient theory of sounds. They distinguished two kinds of intervals

Consonances-sounding well or agreeably together, and Dissonances-sounding ill or disagreeably together, or less agreeably. The consonances were said to be, the unison, octave, fifth, fourth, major and minor third and sixth; all the rest were dissonances.

This distinction is of all things the least essential to mention, since music is in no way simply or chiefly concerned with the diversion or excitement of the senses by pleasant or repulsive sounds. It is much rather the province of music, through the senses, to act on the mind and soul: so that this distinction is merely superficial. An interval, and the sense of perception of it, do not comprehend by far all the requisites wherefrom we should call it beneficial, that is, more simple and easier to be understood. Even in this work, (in the second section of the sixth division,) when better grounded in the science of music, we shall have far different ideas to give. The superficial character of such distinctions is manifest when we see that intervals so different as the fourth, third, and octave, or all the diminished and extreme intervals without exception, are to be all comprehended in one class. In fine, the distinction is quite arbitrary at least in any way in which it might have been made available; for in the perfectly regular progression, 1:2 3:4: 5: 6: 7, there is no more reason to draw a line of separation at 6: 7 than at 5: 6 or 7: 8. And, in fact, some theorists did not stop at that line. Now, they divided the consonances into perfect (the octave and fifth) and imperfect, (the fourth, major and minor thirds and sixth); then, again, the dissonances, into essential and accidental, which last comprised all foreign sounds that entered into a mode or scale. Some, again, on the other hand, have declared the fourth to be a dissonance altogether; while others have considered it to be so only occasionally. Thus they have taken wondrous pains to torment themselves and their scholars, and abandon the essential object of pursuit.

This point and many others of much importance in music are fundamentally discussed in the author's publication, "Die alte Musiklehre im Streit mit unsrer Zeit," at Messrs. Breitkopf and Härtel, in Leipzig. For the convenience of inspection, we have assumed that a sound could be produced by one or two vibrations in a second. This assumption is far from being the fact. The smallest number of vibrations in a second capable of producing a sound, is ascertained to be about 32, and the sound produced is the octave under double C.


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must be -b,

If c-g is to become an extreme fifth, g raised a semitone. Do we wish to convert (a major seventh) into a minor seventh? b must be depressed a semitone to become b. If the minor seventh cb is to become a diminished seventh, b must be again depressed to bb: c-bb is a diminished seventh. In this manner, it is easy to produce minor, diminished, or extreme intervals.

To the beginner, we recommend two practices. First, in writing. Let him sketch out the major intervals from every tone (as we have done at page 17), then all the minor (as we have above). Then let him take first one, then the other major interval, (for example, g-d, gb-db, a major fifth). and convert it into a minor (g-db, gb-dbb), dininished (g-dbb, gb-dbbb),* extreme (g-d, gb-d). Secondly, let him use his faculties in recognising the intervals, particularly the major and minor. Let him seek by the ear the different intervals (for example, major fifths, major and minor thirds and sixths, minor sevenths), from any degree at pleasure. If he think he has hit on them, let him name the tone and measure the quantity of the interval; this will shew him whether he has been successful or not.

It is clear, that with the assistance of the double elevations and depressions, many more intervals might be contrived. The diminished seventh, cbb, might by a further depression of the bb, and by the further raising of the be converted into a doubly and trebly diminished seventh (c-bbb and cx-bob). The extreme fifth, c-g, in like manner, might be twice or three times extended. If it were desirable, indeed, to increase the double sharps and flats, there is no limit to their extension. Fortunately, however, we cannot (as we mentioned in our last note) employ all these easy fabrications in practical art, therefore we will at once lay them aside.

In the previous section we became acquainted with enharmonic tones. We now perceive intervals, which, while both of the same height, are named quite differently, according to the tones on either side of them. They must therefore be called


* Here we meet with a threefold depression, of which we said nothing at page 15. The reason of our silence was, that such depressions or elevations are not in use, or at least, most rare. Such intervals, never scarcely appearing in practice, and displayed only by pedantry, have acquired the inglorious name of paper intervals. They exist, indeed, only on the patient paper, already overburdened enough, even in our days, with useless matter, to the grief of many a weary student. Even our exceptional dbbb above, has been introduced merely to give us the opportunity of cautioning both teachers and scholars against its use, since it is neither necessary nor of any value.

They are easily found, if, in any given interval, one or other of the tones be changed enharmonically. If, for example, in the minor third, c-eb, we change the e enharmonically into d, we produce the extreme second, c-d, whose tones have the same height in our system as the third, c-eb. If in the extreme fifth, c-g, we change the name of the upper tone, we produce the minor sixth, c-ab. In like manner, from the diminished seventh, c-bb, a major sixth, ca or db-bb, would be produced, or also from the fifth, cg, by changing the names of both tones, another fifth, db-ab, would be produced; and other similar changes might be exemplified.

SIXTH SECTION--OF MODES, SCALES, OR SYSTEMS OF DEGREES, USUALLY CALLED KEYS. We have learned that Music has seven degrees, each containing many tones and relations at command. By possibility all the tones and their relations may be brought forward in every piece of music. But as every production of art has a limited sphere of action, and in some sense a determined object, to express a particular and circumscribed range of thoughts and feelings, it is natural, that all tones and their relations should not constantly be used, at least not in the same amplitude and force. On the contrary, every piece of music requires an appropriate selection of tones and their combinations, in which exclusively, or at least principally, the artist should work out his conceptions.

This circumstance also reduces the labour of con

ducting the student in the wide region of musical forms, and prevents his being perplexed and dismayed by their numberless and diversified images. For every composition


present themselves as a foundation. But each one has five forms, and thus furnishes the means of endless varieties of combination. We might begin with c-d-e-f, or c-d-e-f, or cb-d-e-f, or c-d-e-f, c-d-e-f, &c. But from all these possibilities our system has selected two only, as essentially serviceable. We call them THE MODES, SCALES,† OR KEYS, MAJOR AND MINOR. They both agree, in containing the seven degrees. Wherein then do they differ? In the relation which the degrees bear to each other: in the quantities of the intervals formed by each degree, with the first,—

THE SCALES OF THE MAJOR KEYS have major intervals only, between all the other degrees and the first. Therefore the first is followed by a major second, major third, fourth, fifth, sixth

†The succession of tones in which each degree enters only once (or which proceeds in tones or semitones), is called the Diatonic scale. The succession of tones proceding by semitones only, is called the Chromatic scale (page 14). Lastly, that, in which all the tones (at least the chromatic) with their enharmonic double names are introduced (c-eb-e, and so forth), is called the enharmonic scale.

But these two last modes are not calculated to be a foundation for musical composition. It seems even to have been a misunderstanding, to imagine that the ancient Greek musicians made any use of these successions of sounds, or anything resembling them, as modes or grounds of musical construction; although the assertions of the Greek theorists sustain that opinion (see the author's article on Gr. Music, in the Universal Lexikon der Tonkunst). From those theorists the idea has descended to us. There remains, however, but one thing to say of these three pretended Modes, which is, that the whole arrangement in this view is utterly worthless.

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