We typically sense harmonic progressions with root-position chords as being stronger than those in which one or both chords are in inversion. Therefore, we would expect that a value which measures harmonic attraction would be higher when applied to progressions involving root-position triads than when applied to progressions where one or both chords are in inversion. However, the formulas used to calculate harmonic attraction in Fred Lerdahl's Tonal Pitch Space (2001) do not take into account inversion; rather, Lerdahl accounts for the differences we hear between inverted and root-position triads as a measurement of surface dissonance. Therefore, the application of harmonic attraction does not accurately reflect the differences that we hear in progressions between root-position triads and triads where one or both chords are in inversion.
This paper will demonstrate how the dominant chord in inversion can be perceived as having an alternate pitch as its root other than the traditional pitch. The perception of a different pitch class as the root of a chord can affect our perception of root movement in a progression, thereby altering our measurement of chordal distance. Revising the difference in chordal distance between perceptual roots and traditional roots with Lerdahl's harmonic attraction formula gives us a new formula that successfully matches our intuitions regarding our perception of inverted dominant triads in dominant-tonic progressions as being harmonically weaker than when both the dominant and tonic triads are in root position.