Richard Parncutt. (1989). Harmony: A Psychoacoustical Approach. Berlin: Springer-Verlag.

Parncutt proposed a model of harmony perception based on Terhardt's theory thaat tonal meaning derives from psychoacoustic processes involved in the perception of "virtual pitch." Whereas Terhardt's musical concerns were mostly with chord roots, Parncutt wants to extend the theory to explain successions of chords in terms of the smoothness of motion between virtual pitches. A model is proposed with free parameters corresponding to listening strategies. Experiments are conducted to estimate reasonable values for these parameters. Finally, the model is tested on several theoretical and applied cases, and found to make sensible and interesting predictions about sonorities and chord progressions.

I. Background.

Whereas Rameau focused on physical properties of resonating bodies, Terhardt focused on the familiarity of the auditory system with the sound of resonating bodies. This consitituted a shift from physics to perception.

II. Psychoacoustics.

Spectral pitch is the pitch of a pure tone sensation. Virtual pitch is the pitch of a complex tone sensation, and is normally ambiguous. A virtual pitch results from the spontaneous recognition of the familiar pattern of spectral pitches of a harmonic complex tone.

III. Psychomusicology.

• 3.1. Conditioning. Human fetuses can hear at 20 weeks before birth. In sheep, fetus heartbeats change rate at unfamiliar sounds. Most sustained pitches heard at this age (or any age) are voices. According to Terhardt, pitch recognition is learned from listening to voices.
  
  
• 3.2. Consonance.
  
  3.2.2. Roughness is the interaction between proximal pure tones. Pure tonalness is a measure of the number of clearly audible pure tones. Complex tonalness is the clarity of the single virtual pitch evoked by a chord. Complex tonalness is enhanced when complex pitches are arranged along the harmonic series of the complex virtual pitch. Roughness and tonalness are negatively correlated. Roughness depends mostly in interaction between equally loud pure tones, so playing one note in a chord louder can reduce roughness.
  
  3.2.3. Pitch commonality can be defined as the degree to which two chords have (virtual) tone sensations in common. Pitch distance is a measure of distance between the tone sensations in two chords. Both of these are hypothesized to measure to measure the joint consequence of sequential tones.
  
  
• 3.4. Tonality.
  
  3.4.1. According to Terhardt, the root of a chord is its complex virtual pitch. In general, minor triads have weaker roots.
  
  3.4.3. The tonic of a scale is the root of a chord formed by scale notes not involved in tritone relationships.
  
  3.4.4. The dissonance and root ambiguity of the minor triad may have contributed to its emotional meaning.

IV. Model.

• 4.1. General aspects.
• 4.1.2. There are 4 free parameters in the model, indicating the degree of analytic listening with regard to:
  1. effects of masking - k _M
  2. spectral pitch vs. complex pitch - k _T
  3. individual complex pitches vs. sonority - k _S
  4. pitch commonality vs. pitch proximity - k _R
• 4.1.3. Spectral tones are converted to equal-temperament. (This facilitates later statistical calculations.) This step would create pitch-shifting in Terhardt's model, but is irrelevant for musical contexts, which are categorically perceived.
• 4.2. Input. Harmonics 1-16 are included in each tone.
• 4.3. Masking. Masking is incorporated to simulate the interaction of pure tones.
• 4.4. Recognition of harmonic pitch patterns. See figure 4.3.
• Salience. Within a chord, the following properties are calculated: the estimated multiplicity of heard tones, the salience of each tone (taking into account the multiplicity of tones), and the salience of each pitch chroma.
• 4.6. Sequential pitch relationship.
• 4.6.1. Pitch commonality between two chords is calculated roughly as the correlation between the pitch salience of two chords across the frequency spectrum.
• 4.6.2. Pitch distance between two chords is calculated roughly as the difference between covariance in pitch salience (weighted by distance) and the product of the individual weighted variances.

V. Experiments.

• 5.2. Multiplicity. An experiment is conducted to determine how many pitches listeners can hear within sonorities. See figure 1. The experimental question: "How many tones do you hear in the sound?" The results allow values to be estimated for k _M and k _S.
• 5.3. Pitch weight. A complex sound is presented, followed by one of its tone components. The question: "Does the tone sound like it is part of the first sound?" From this, values for k _T and k _R can be estimated.
• 5.4. Similarity of piano tones. This was a first test of the sequential components of the model. The question: "Rate the similarity of the two tones on a 4-point scale."
• 5.5. and 5.6. A continuation of the same experiment with synthetic tones.
• 5.7. Similarity of chords. See figure 5.7.
• 5.8. The final summary of parameter fits is in Table 5.1.

VI. Aplications.

• 6.1.3. Multiplicity (denumerability?) of pitches can be calculated. See figure 6.1.
• 6.1.4. Tonalness (complex tone audibility, i.e. consonance). See figure 6.2.
  • 2-part writing: for homogeneity of consonance, avoid m2, M7, P8
  • 3-part chords: 5th/4th > tritone/m6, major > minor, root position > inversions (except diminished)
  • Augmented triad has fine tonalness, but an ambiguous root.
  • Best tetrad is m65. For others seventh chords, root position is more tonal. D7 has the strongest root.
  • Close spacing in low registers and wide in highr result in low tonalness.
  • Root doubling produces best tonalness. 3rd doubling in major and 5th doubling in minor is worse.
• 6.1.5. Roots of simultaneities can be calculated using octave spaced tones.
  • Only two intervals are clearly rooted: M3 and P4.
  • The major triad is clearly rooted. The minor triad also implies the third as root.
  • Diminished triad also implies the root of a D7 chord.
  • m7 implies its third as root -- a 6th chord? The same is true for the half-diminished 7th, but D7 is also implied.
  • M7 has both root and third as roots. Both this and the m7 may be the effects of octave masking between root and seventh.
  • Diminished 7th implies all chordal-notes as roots, as well as semitones below each. Adding one of these notes makes it a D9.
• 6.2. Progression.
• 6.2.1. Pitch commonality.
  • Interval pitch commonality, from lowest to highest: (Fig. 6.4) P8, P5, M2, M3, P4, m7, m3, M6, TT, m6, m2, M7. Compare to circle of fifths: P8, P5/P4, m7/M2, M3/m6, m2/M7, TT
  • Semitone parallel motion of dyads: P8/P5, M6, P4, m3/m7, M2, m6/M3, TT/M7, m2
  • Commonalities are computed for successive triads. Table 6.3.
• 6.3. Analysis. Figures 6.7-6.8.

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