Revision of Terhardt's Psychoacoustical Model of the Root(s) of a Musical Chord

Abstract

The predictions of Terhardt's octave-generalized model of the root of a musical chord occasionally disagree with music theory (notably, in the case of the minor triad). The model is improved by assigning appropriate weights to the intervals used in the model's "subharmonic matching" routine. These intervals, called "root-supports," include the P8 (unison), P5, M3, m7, M9 (M2), and m3. The new model calculates the salience of each pitch class (C, C#/Db..B) as an absolute value. The most likely candidate for the root of a chord corresponds to the most salient pitch class in all cases where the root is unambiguously defined in music theory. The model also calculates a "root ambiguity" value for each chord, a measure of its dissonance. Effects of voicing (inversion, spacing, and doubling) and context on the root are considered.

Commentary on Cook & Fujisawa's "The Psychophysics of Harmony Perception: Harmony is a Three-Tone Phenomenon"

“…This is consistent with my approach in Parncutt (1988Parncutt ( , 1989, as well as for example Hutchinson and Knopoff (1978), whose algorithm can be applied to any sonority of any number of notes or partials. But the above arguments suggest that the situation may be more complex: the consonance of a chord type may, in fact, depend directly on both the chord as a whole and the individual intervals.…”

“…Clearly, there is something wrong with my model of root ambiguity: both the octavegeneralised version (Parncutt, 1988) and the non-octave-generalised version (Parncutt, 1989) predict incorrectly that the root of an augmented triad is less ambiguous than the root of a diminished triad (i.e. that the augmented triad is more consonant in this respect).…”

“…This also explains for example why the Renaissance-Baroque compositional practice of tierce de Picardie involves a major triad following a minor passage -and not the other way around. The pitch salience profiles of Parncutt (1988) suggest that the third of a minor triad competes with the (conventional) root for the status of (perceptual) root; similarly, the key profiles of Krumhansl & Kessler (1982) suggest that the third scale member of a minor key competes with the (conventional) tonic for the status of (perceptual) tonic.…”

“…Of all 19 possible triad types in the chromatic scale (12 pitch class sets of cardinality 3, plus the inversions of those 7 that are not symmetrical; Rahn, 1980), only two contain a perfect fifth (or fourth) and no seconds: the major and minor triads. I made this apparently trivial observation in Parncutt (1988). Perhaps it is not so trivial after all, because it answers basic questions such as "What is so special about the major and minor triads?"…”

“…In both cases, we may safely assume that all listeners experience the sensation of the roughness, but that that sensation is evaluated differently depending on the musical or social context and the listener's cultural background. b. I introduced the term root ambiguity in Parncutt (1988Parncutt ( , 1989. A chord with a clear root tends to have a clear function in a harmonic progression, so one may assume that chords with clear roots are (or were) preferred by composers of tonal music.…”