An inverted chord is a chord which has a note other than its root note as the bass note. Since Rameau (1722), chords remain equivalent when inverted, being functions rather than sonorities. However, before Rameau factors including the regola delle terze e seste, "rule of sixths and thirds", which required the resolution of imperfect consonances to perfect ones, precluded the analysis of 64 sonorites as inversions of 53 sonorities.
For example, the root position of a triad of C major has the C in the bass:
A triad in root position, therefore, consists of the root note and a third and a fifth above it. Triads in root position are also in normal form.
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 consists of the root plus a third and a sixth above it. The second inversion has the fifth, the G, in the bass:
A triad in second inversion, therefore, consists of the root plus a fourth and a sixth above it.
The third inversion of a triad does not really make much sense to discuss, since inverting the second inversion 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 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.
Notations for inverted chords
There are at least four different notations for the inversions of chords.
(i) Perhaps the most commonly used method is figured bass. In this system, first inversions are normally indicated by the digit 6 and second inversions by the digits 64. A full presentation of figured bass notation is given in the Wikipedia article on this subject.
(ii) The letters a, b, c, etc., may be placed after any chord symbol to indicate the root, first and second inversion respectively. Hence the C chord below, in first inversion (i.e. with E in the bass) may be notated as Cb. (If no letter is added, the chord is assumed to be in root inversion, having the same meaning as if 'a' had been added explicitly.)
(iii) A less common, but occasionally used, notation for chord inversion is to place the number 1, 2 or 3 etc. after a chord to indicate that it is in first, second, or third inversion respectively. Hence the C chord above, in first inversion (i.e. with E in the bass) may be notated as C1. No number is added in the case of a chord in root inversion. This notation should not be confused with a quite different meaning of the same notation, where a number is placed after a note name to indicate the octave in which a single note is to sound, e.g. C4 is often used simply to mean the single note middle C.
(iv) A notation for chord inversion often used in popular music is to write the name of a chord, followed by a forward slash, and then the name of the note that is to sound in the bass. For example, the C chord above, in first inversion (i.e. with E in the bass) may be notated as C/E. Interestingly, this notation works consistently even when a note not present in a triad or other chord is to sound in the bass, e.g. F/G is a way of notating a particular approach to voicing a G13th chord. This should not be confused with notations of the "function of function" style, for instance the subdominant of the dominant is IV/V or S/D. (Courtesy of Wikipedia)