Ernst Terhardt (1974). "Pitch, consonance and harmony." Journal of the Acoustical Society of America, Vol. 55, No. 5, pp. 1061-1069.

I. The Problem of Consonance.

• Certain musical intervals are considered consonant, regardless of their pitch height.
• Modern theories of sensory dissonance imply that:
  1. Intervals with simple interval ratios should sound dissonant when low in pitch.
  2. Supposedly "dissonant" intervals should sound consonant when high or widely spaced.
  3. Even single tones can sound dissonant.
• If intervals and harmonics are real in the sense in which they are conventionally understood, some other principle must be in play.
• Rameau hypothesized that chords have a single tonal center, and Riemann hypothesized that chords produce unifying subharmonics.

II. Virtual Pitch.

A. General principles of pitch perception.

• Two modes of hearing pitch are proposed: analytic mode and synthetic mode, which result in spectral (pure tone) pitch and virtual pitch, respectively.
• Virtual pitch is a learned method of integrating pure tones into implied pitch, much like implied contours can be seen in vision.

B. Pitch perception in the synthetic mode.

• The model of synthetic mode first produces a set of virtual pitch frequencies and their probabilities of being detected. Next it selects the pitch with the highest probability (see figure).
• The probability of a virtual pitch is determined by the strength of the match between its implied harmonic series (the learned pattern) and the audible pure tones in the sound.
• Spectral information is assumed to be processed peripherally. Virtual information is assumed to be processed centrally.

C. Music perception.

• Octaves sound similar because most virtual pitches are octave spaced.
• A major chord has pronounced virtual pitches one and two octaves below the root. This gives a major chord its tonal meaning.

III. Conclusions.

• Musical sounds are govered by two principles:
  1. The principle of minimum roughness, and
  2. The principle of tonal meanings
• Just tuning does not maximize the virtual pitch roots of chords. Every tuning is a compromise, according to this model.
• Other synthetic sonorities not based on harmonic pitches may also produce strong tonal meaning.

• Harmony & Voice-leading Course Page