How this calculator works
A chord is defined by its pitch classes, and the bass note can be any of them. Moving the lowest note up an octave produces an inversion without changing the chord. C major is C, E, G in root position. The first inversion puts E in the bass with the notes E, G, C; the second inversion puts G in the bass with G, C, E. Inversions change the colour and the voice leading rather than the harmony, and bass movement between inversions is often smoother than jumping between root positions.
Formula
for a chord with n tones, inversion k takes the tones in cyclic order starting from tone k, and each tone that wraps past the end is raised by twelve semitones. The bass note is the k-th chord tone. Root position is k = 0, first inversion k = 1, second k = 2 and third k = 3.
Worked example
C major has the tones C, E and G. First inversion starts on E and raises the wrapped C by an octave, giving E, G, C with E in the bass. Second inversion gives G, C, E with G in the bass. G dominant seventh, G B D and F, has a third inversion: F, G, B, D, with F in the bass.
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Frequently asked questions
Does an inversion change the chord name?
The chord stays the same; only the bass changes. In notation the inversion is written as a slash chord, such as C/E for C major in first inversion, read as C major over E. Players use the bass note to pick a voicing, not to rename the harmony.
Why use inversions at all?
They keep the bass line moving by step instead of leaping, and they let several chords share notes, which sounds smoother. A common progression such as C, G/B, Am, F uses a descending bass line that would be more jumpy with root-position chords.
What is the highest inversion?
A three-note triad has three positions, so the second inversion is the highest. A four-note seventh chord adds a third inversion. Extended chords have more notes and more inversions, but players often leave tones out, so the count depends on the voicing.
Do the octaves matter?
This page shows the order of notes from the lowest upward and puts wrapped tones an octave higher. Actual players choose octaves freely to fit the instrument and the arrangement. The chord tones and the bass note are what define the inversion.