Inversional equivalency
In music theory, the word inversion has several meanings.
Inverted chords
An inverted chord is a chord which has a note other than its root in the bass. For example, the root position of a triad of C major has the C in the bass:
A triad in root position, therefore, is made up of the root note and a third and a fifth above it.
The first inversion of the same triad has the E, the third of the triad, in the bass:
This means that a triad in first inversion is made up of the root plus a third and a sixth above it. In figured bass, a first inversion is indicated by the number 6.
The second inversion has the fifth, the G, in the bass:
A triad in second inversion, therefore, is made up of the root plus a fourth and a sixth above it. The figured bass notation for this is 64.
The third inversion of a triad does not really make much sense to discuss, since inverting the second version just leads to the tonic again, an octave higher. Chords of four notes or more, however, can be in their third inversion: the third inversion of a dominant seventh in C major, for example (made up of the notes G, B, D and F) has the seventh, F, in the bass. This gives a chord made up of the root plus a second, fourth and sixth above it. The figured bass notation is accordingly 642.
The terms "root", "first inversion", and "second inversion" may also be applied to chords in which the notes are not closely spaced. For instance, C-G-E, where the E is a major sixth above G, is also considered to be in root position, and more generally, any C major chord in which C is the lowest note is considered to be in root position. Similarly, any C major chord with E on the bottom counts as a first inversion, any C major chord with G on the bottom counts as a second inversion; and analogously for all other chords.
Inverted melodies
When applied to melodies, the inversion of a given melody is the melody turned upside-down. For instance, if the original melody has a rising major third (see interval), the inverted melody has a falling major third. Similarly, in twelve-tone technique, the inversion of the tone row is the so-called prime series turned upside-down.
Inverted intervals
An interval is inverted by raising or lowering either of the notes the necessary number of octaves, so that both retain their names (pitch class) and the one which was higher is now lower and vice versa. For example, the inversion of an interval consisting of a C with an E above it is an E with a C above it - to work this out, the C may be moved up, the E may be lowered, or both may be moved.
Under inversion, perfect intervals remain perfect, major intervals become minor and the reverse, augmented intervals become diminished and the reverse, seconds become sevenths and the reverse, thirds become sixes and the reverse, and fourths become fifths and the reverse. Thus a perfect fourth becomes a perfect fifth, an augmented fourth becomes a diminished fifth, and a simple interval (that is, one that is narrower than an octave) and its inversion, when added together, will equal an octave. See also complement.
Inversion in counterpoint
When applied to counterpoint, a contrapuntal inversion of two melodies simultaneously being played by two voices is the switching of the melodies between voices, so that the upper voice melody is now played in the lower voice, and vice versa.
Inversional equivalency
Inversional equivalency is the concept that intervals, chords, and other sets of pitches are the same when inverted. It is similar to enharmonic equivalency and octave equivalency and even transpositional equivalency. Inversional equivalency is used little in tonal theory, though it is assumed a set which may be inverted onto another are remotely in common. However, taking them to be identical or near-identical is only assumed in musical set theory.
Inversion in musical set theory
In musical set theory inversion may be usefully thought of as the compound operation transpositional inversion, which is the same sense of inversion as in the paragraph above, with transposition carried out after inversion. Pitch inversion by an ordered pitch interval may be defined as:
which equals
First invert the pitch or pitches, x=-x, then transpose, -x+n.
Pitch class inversion by a pitch class interval may be defined as:
For other meanings of the word inversion, see inversion.
|
