Since hundreds of years, composers and theorists of music have been aware that each individual sound of music, especially chords, bears a close relationship to a fundamental note that characterizes and determines its harmony. In the 16th Century, Generalbass notation occurred, i.e., a reduced notation system that takes advantage of the fact that the harmonic layout of a piece of tonal music can be prescribed by just denoting the corresponding sequence of bass notes. Primarily, a bass note indicated a major triad of which that note was the "root". Variants thereof, i.e., minor chords, sixths, diminished chords, etc., were indicated by numbers added to the bass notes. Until present days, this kind of simplified notation is widely in use, essentially in improvised music such as in Jazz.
While in the Generalbass technique the performer's task was to find appropriate chords to given bass notes, a crucial task of harmonic analysis of any given piece is to find the fundamental notes of the prescribed sounds. Indeed, many relevant musical features of a given chord can be characterized by a single note, i.e., the chord's root. There is a mutual relationship between chords and fundamental notes, i.e., roots.
In many types of music the fundamental notes are not necessarily played on any instrument. Rather, they are implied in both the "horizontal" (melodic) and "vertical" (harmonic) structure of the music. In this sense they are virtual. Countless sophisticated examples of this phenomenon are included in J. S. Bach's music. Another, fairly simple example is the "Alberti" accompaniment of melodies that so frequently was employed by W. A. Mozart, e.g., in his piano sonatas. Those "Alberti" notes are not to be regarded as bass notes (comparable to those of the baroque Generalbass music) but as arpeggized chords that in turn represent virtual bass notes.
So it is more or less evident that, from the beginning of polyphony, tonal music has been based upon, and has taken advantage of, the mutual relationship that exists between chords and fundamental notes - just as if that relationship were a basic auditory phenomenon such as the affinity of tones, in particular, octave equivalence.
J.P. Rameau (1722, 1750) was the first who, in his Traite de l'Harmonie, explicitly treated the fundamental notes as an auditory percept. He regarded the basse fondamentale as an "implied" or "inferred" auditory feature of sound which nevertheless has psychological reality and is of crucial significance in tonal music.
And this is exactly what the basse fondamentale actually is: An auditory percept, though a virtual one. Any pitch corresponding to any fundamental note merely is a virtual pitch  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  p. 398, 404. So, the theory of virtual pitch can be employed as a universal tool to determine the roots of any type of musical sound  (see topic harmony).
This implies that the root of musical chords is not merely a theoretical (though quite useful and smart) concept, but that it is an attribute of auditory sensation, i.e., virtual pitch. It may be true that in a typical musical context one is rarely, if ever, aware of perceiving roots as definite pitches. This does not disprove, however, that those virtual pitches are perceived unconsciously, and that they may become conscious when attention is properly guided. Apparently, Rameau was the first who became aware of those "unconscious" pitches, and made them apparent in his concept of harmony.
In fact, even for ordinary listeners those pitches can be lifted into conscious perception. An efficient trick to accomplish this is designing a fairly rapid sequence of chords in random inversions, the roots of which define a melody. Provided that the chords are designed such that this melody is not discernible from the tones of the chords, presentation of the sequence to listeners is a test of whether or not they can perceive the virtual pitches of the roots. If they can recognize the melody "hidden" in the chord sequence, one can conclude that they have perceived a corresponding sequence of virtual pitches ,  p. 28 (cf. the CD attached to ). Through many years I have presented this demonstration to many audiences (The melody was the French tune "Sur le pont d'Avignon"); throughout, 80-90% of the listeners recognized the melody.
In 1976/77 we have carried out a formal test of auditory root recognition. Students of music (who already had taken some lessons on transcription of music) were asked to transcribe the virtual pitches they heard in brief sequences of 5 sounds. These sounds were either dyads or triads designed such that the pertinent root sequences were composed of the tones C, D, E, F, G. Both the composition of the dyads and triads, respectively, and the order of the five root notes were randomized. In this test, three out of nine subjects achieved 70%-83% correct recognition of root sequences. Another four subjects scored 8%-20%, and two subjects scored just above the borderline of significance (Note that there are 5! = 120 permutations of the above five root notes. So the chance for correct guessing without recognizing anything was only 0.8 percent.) ,  p. 400.
In summary, there can be hardly any doubt that Rameau's basse fondamentale, i.e., the root of chords, has got its psychophysical explanation in the above findings.
Author: Ernst Terhardt firstname.lastname@example.org Feb 28, 2000