INVERSION OF INTERVALS.
Intervals are said to be inverted when the lower note of the original interval is placed an octave higher, thereby becoming the upper note of the interval thus formed. Example: the inversion of
is
. The same letters are in both intervals, but the first interval is a third and the inverted interval is a sixth.
Rule 8. The Sum of an Interval and Its Inversion is Nine.
The above rule, therefore, gives the following inversions:—
| a prime | inverts | to | an octave | (1 + 8 = 9) |
| a second | " | " | a seventh | (2 + 7 = 9) |
| a third | " | " | a sixth | (3 + 6 = 9) |
| a fourth | " | " | a fifth | (4 + 5 = 9) |
| a fifth | " | " | a fourth | (5 + 4 = 9) |
| a sixth | " | " | a third | (6 + 3 = 9) |
| a seventh | " | " | a second | (7 + 2 =9) |
| an octave | " | " | a prime | (8 + 1 = 9) |
To find to what intervals ninths, tenths, elevenths, twelfths, etc., invert, consider them respectively as seconds, thirds, fourths, fifths, etc., and consider the lower note placed two octaves higher instead of one octave.
Qualifications invert in the following manner:—
| major | intervals | invert | to | minor | intervals |
| minor | " | " | " | major | " |
| perfect | " | " | " | perfect | " |
| diminished | " | " | " | augmented | " |
| augmented | " | " | " | diminished | " |
By the use of the above table and rule 8, all inversions may be determined. Examples:—
| major 2nd | inverts to | minor 7th |
| major 6th | " " | minor 3rd |
| perf. prime | " " | perf. 8th |
| perf. 4th | " " | perf. 5th |
| aug. 4th | " " | dim. 5th |
| dim. 7th | " " | aug. 2nd |
| etc. | etc. |
The prime is also called an unison, but in speaking of intervals, it should always be called a prime. Correctly speaking, a perfect prime is not an interval, but in the theory of music it is so called. There is good reason for making this error, but none for calling a diminished prime a diminished unison. Notice that the diminished second as well as the perfect prime is an unison.
Intervals are considered both harmonically and melodically, or in other words, both when sounded together or separately. In either case, the lower note is the one from which to determine the interval.
CHORD CONSTRUCTION.
A chord is a combination of two or more tones sounded simultaneously. All chords are constructed in thirds. The fundamental tone of a chord is the tone on which the chord is constructed.
A chord of three tones is a triad which consists of a fundamental together with its third and its fifth. Triads are divided into two classes, independent and dependent. The independent triads have no dissonant intervals and may end a composition. The dependent chords have one or more dissonant intervals and are "restless" chords and demand another chord to follow. The progression of a dependent chord to an independent chord, thereby obtaining a restful effect, is called resolution.
There are two kinds of independent triads, major and minor. A major triad consists of the fundamental, the major third, and the perfect fifth. Example:—
. A minor triad consists of the fundamental, the minor third, and the perfect fifth. Example:—
.
If the fifth of a triad is augmented or diminished, the triad is a dependent. Dependent triads are found constructed on the subtonic of major keys; on the subtonic, supertonic, and mediant of the minor keys. The triad on the mediant of the minor key is an augmented triad and the first three mentioned triads are diminished.
Music written for four voices necessitates the doubling of one of the factors of the triad. Any factor of the triad may be doubled. The factor most frequently doubled is the fundamental in the octave, double octave, or unison. The four voices are soprano (high female voice), alto (low female voice), tenor (high male voice), and bass (low male voice). Chords are figured under the bass by Roman numerals. Large numerals designate major triads; small numerals designate minor triads; large numerals with the mark (´) affixed designate augmented triads; small numerals with a cipher affixed designate diminished triads. The notation of the triads on each degree of the major and minor scales follows:—
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N. B. Although doubled in the above examples, the fundamental of the subtonic triad is seldom doubled in four voice writing and if doubled, a bad progression results in many cases.
The three upper voices in a fundamental chord may be arranged in a different manner:—
With the fundamental in the soprano:—
.
With the third in the soprano:—
.
If the three upper voices do not exceed the compass of an octave, the chord is said to be in "close position." If the three upper voices exceed the compass of an octave, the chord is said to be in "open position."
INVERSION OF TRIADS.
A chord is inverted when a factor other than the fundamental is in the bass. The first inversion of the triad is where we have the third in the bass. It is called the chord of the sixth, because the fundamental is the sixth of the bass. This chord is figured by a small Arabic figure (⁶) over the bass note, the Roman numeral under the bass showing the fundamental. Examples:—
[[Midi]] [[audio/mpeg]] [[XML]]
The second inversion of the triad has the fifth in the bass and is called the chord of the sixth and fourth, or six-four chord, because the fundamental is the fourth of the bass. This chord is figured by the small Arabic figures (64) over the bass, the Roman numeral under the bass showing the fundamental. Examples:—
[[Midi]] [[audio/mpeg]] [[XML]]
THE SEVENTH CHORD.
The seventh chord is obtained by adding the seventh of the fundamental to any triad. A seventh chord may be formed on each degree of the major and minor scales. It is figured with the Roman numerals below the bass and a small Arabic (7) over the bass. The notation of the seventh chords on each degree of the major and minor scales follows:—
C Major
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a̲ Minor.
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The tonic and subdominant of the major mode and the submediant of the minor mode are formed with the major triad and the major seventh. The dominant seventh in both modes is formed with the major triad and the minor seventh. The seventh chords on II, III and VI in the major mode and on IV of the minor mode are formed with the minor triad and the minor seventh. The seventh chords on VII° in the major mode and on II° in the minor mode are formed with the diminished triad and the minor seventh. The subtonic seventh chord in the minor mode is called the diminished seventh and is formed with the diminished triad and the diminished seventh. The seventh chord on the mediant in the minor mode is formed with the augmented triad and the major seventh. The seventh chord on the tonic of the minor mode is formed with the minor triad and the major seventh. In four voice writing, all the seventh chords with the exception of those on the subtonic of both modes are often written without the fifth and with the doubled fundamental. All seventh chords are dependent chords and their natural resolution is to the chord the fundamental of which is situated a fourth above or a fifth below the fundamental of the seventh chord. This progression is called "cadencing resolution." The subtonic seventh chord of both modes may also naturally resolve to the tonic. The most important seventh chord is the dominant, which resolves to the tonic. This progression is called the authentic close. Another method of ending a composition is by the plagal close which is a progression from subdominant harmony (triad) to tonic. The plagal close is preceded by the authentic close and is also called the after cadence and the Amen cadence.
INVERSION OF SEVENTH CHORDS.
The first inversion of the seventh chord is called the chord of the fifth and sixth (six-five chord). The chord is figured by the Roman numeral below the bass note and the Arabic figures (65) over the bass. Example:—
The second inversion is called the chord of the third, fourth and sixth (four-three chord). It is figured by the Roman numeral below the bass and the Arabic figures (643) or simply (43) over the bass. Example:—
The third inversion is called the chord of the second and fourth. It is figured by the Roman numeral below the bass and the Arabic figures (642), (42), or simply (2) over the bass. Example:—
The study of chord progression, altered chords, melody writing, passing tones, etc., belongs properly to the study of harmony and counterpoint which is not the subject of this volume.