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flat, next below, which is composed of eight ehromatic degrees;

2. The perfect fourth, as C natural, F natural, composed of five chromatic degrees, produces, by inversion, the perfect fifth, as C, and F, next below, which is composed of seven chromatic degrees;

3. The augmented fourth, as C natural, F sharp, next above, composed of six chromatic degrees, produces, by inversion, the diminished fifth, which is also composed of six chromatic degrees.

Intervals of fifths, of which there are three species, produce, by inversion, fourths of the same number and species; thus

1. The diminished fifth, as C natural, G flat, next above, composed of six chromatic degrees, produces, by inversion, the augmented fourth, as C natural, and G flat, next below, which, as before stated, is composed of six chromatic degrees;

2. The perfect fifth, as C natural, G natural, next above, composed of seven chromatic degrees, produces, by inversion, the perfect fourth, as C natural, G natural, next below, which is composed of five chromatic degrees;

3. The augmented fifth, as C natural, G sharp, next above, composed of eight chromatic degrees, produces, by inversion, the diminished fourth, as C natural, and G sharp, next below, which is composed of four chromatic degrees. Intervals of sixths, of which there are four species, produce, by inversion, thirds of the same number and species, thus

1. The diminished sixth, as C sharp, A flat next above, composed of seven chromatic degrees, produces, by inversion, the augmented third, as C sharp, A flat, next below, which is composed of five chromatic degrees:

2. The minor sixth, as C natural, A flat, next above, composed of eight chromatic degrees, produces, by inversion, the major third, as C natural, A flat next below, which is composed of four chromatic degrees:

3. The major sixth, as C natural, A natural, next above, composed of nine chromatic degrees, produces, by inversion, the minor third, as C natural, A natural, next below, which is composed of three chromatic degrees.

4. The augmented sixth, as C natural, A sharp, composed of ten chromatic degrees, produces, by inversion, the forbidden interval of the diminished third, as C natural, A sharp, next below, which is composed of two chromatic degrees. Intervals of sevenths, of which there are three species, produce, by inversion, seconds, thus

1. The diminished seventh, as C sharp, B flat, next above, composed of nine chromatic degrees, produces, by inversion, the augmented second, as C sharp, and B flat, next below, which is composed of three chromatic degrees;

2. The minor seventh, as C natural, B flat, next above, produces, by inversion, the major second, as C natural, B flat next below, which is composed of two chromatic degrees.

3. The major seventh, as C natural, B natural, composed of eleven chromatic degrees, produces, by inversion, the minor second, which is composed of one chromatic degree.

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Minor and major thirds and sixths, in the composition of melody or harmony, being of themselves either way satisfactory to the ear, which is the umpire of all musical sounds, are distinguished by the appellation of imperfect consonances; fourths, fifths, and octaves, are denominated perfect consonances, because, by the alteration of either of them by a sharp or flat, they are immediately rendered unsatisfactory to the ear. These and all other chromatic intervals are dissonant, as is the case with seconds and sevenths; and the fourth of the diatonic scale when forming an integral part of the dominant harmony of the seventh; and the sixth, when forming that of the dominant ninth, are also regarded as dissonances requiring to be regularly resolved into the perfect harmony of the Tonic, thus

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Upon these simple combinations of different sounds, the first of which must be considered the cause, and the last the effect of that cause, the whole science of harmony depends: the powers of the first providing, by the application of a flat, or a sharp, taken in totality, or in part, for every species of harmonic combination used in the present day; as the materials of the latter provide, by the occasional application of a flat to its third note, every sound proper for the resolution of the dissonances arising, directly, or indirectly, from the harmony of the dominant ninth. Thus the lowest note of the foregoing example of the harmony of the dominant ninth, called the dominant root, accompanied only by the major third, and fifth, thus-G BD, affords the modes of the tonic major, perfect harmony CEG; but although the materials of which these intervals are composed be absolutely the same, their effects as employed in succession are of a widely different nature, the first governing entirely the key note of the second which is C, the key note of the passage, thus—

1 2 3 4 5 6 7 8 8 765 4 3 2 1 consisting of five tones and two half tones, the latter falling between the third and fourth, and seventh and eighth intervals. We postpone other considerations of this diatonic scale till we have developed the whole of the harmonic powers of the dominant ninth.

The application of the sign of a flat to the third of the dominant root, thusforms the model of the perfect minor harmony; and the same union of different sounds, varied by the addition of another flat, constitutes the softest of all discords, which is the harmony of the minor third and diminished fifth; the acute, as well as the middle, sound forming a dissonance, both of which must be regularly resolved, and in the following manner, viz.

It follows therefore of consequence that every note composing the dominant harmony of the ninth must be considered and treated as a dissonance, excepting the root G: the A cadencing or falling upon G, the F upon E, the D upon C, the B cadencing or rising to C, i. e. from necessity, or according to the principles as established by nature: the G, which is common to both harmonies, forms the link by which they are united. From this simple process, of which the tonic, or key note, C, may be said to be the centre of gravity, upon which all other sounds resolve, we obtain the model and origin of the twelve major diatonic scales; all other major scales being only transpositions of this natural, or primary order of sounds, viz.

Varying this third dominant combination of sounds, i. e. by placing the sign of a sharp to the highest sound, and omitting the two flats, we have the harmony of the major third and augmented fifth, when the fifth forms the principal dissonance, requiring, together with the third, resolution into perfect harmony, thus, From the harmony of the dominant ninth we have therefore derived four species of combinations of thirds and fifths. From the same source we also derive four species of the harmony of, or belonging to, the seventh viz.

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Proceeding still by the system of thirds, we obtain the dominant major ninth composed of the major third, perfect fifth, minor seventh, and major ninth, all of which are dissonant, as before explained (see page 291.) By the application of a flat to the ninth interval of this harmonic combination, we have the harmony of the dominant minor ninth, composed of the major third, present fifth, minor seventh, and minor ninth, the whole of which sounds are also dissonant, and resolve in the following manner, viz.

and the highest note ascends as usual, thus-
Gis the root of the discord
still,though D flat is termed
the bass, because sung by
the deepest voice, or played
by a bass instrument; the

second combination consists of the harmony of
the major third, perfect fifth, and augmented
sixth, when all intervals are dissonant, resolving
sooner or later into perfect harmony, thus-
i.e.one after another, for
reasons hereafter to be
explained. It will be
seen that this combina-
tion of dissonant sounds

is the same as the dominant minor ninth, with the exception of having the perfect fifth one degree, and omitting the root. The last harmonic combination of sounds, making the thirteenth, is expressed in the following manner, consisting of the union of the major third, minor seventh, and augmented fifth, and resolving thusIt must be observed that the resolutions of the various dissonant harmonies, arising from the dominant ninth, may be immediately effected

in minor as well as major perfect harmonies, with the exception of Nos. 4, 8, and 13. The dissonances, therefore, of No. 11, may also resolve in the following manner, viz.—

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From the harmony of the dominant minor ninth are derived two other combinations of sounds; the first is composed of the harmony of the major third, augmented fourth, and augmented sixth, which is effected by lowering the perfect fifth one-half tone, and omitting the note constituting the minor ninth, when the two gravest notes, as dissonances, descend one degree each,

Remarks upon the respective powers of the foregoing dissonant and consonant harmonies, and of the situations they occupy in the diatonic major and minor scales.

It has undoubtedly been remarked that the notes composing the several tonic harmonies, introduced for the proper resolution of the dis

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