Tone of Voice and Mind : The Connections between Intonation ...

268 Appendix 2
The second point is that the µ A factorinEq.A2-3isincludedexpressly
for the purpose of obtaining a higher instability value for the augmented than
the diminished chords (Tables A2-1 and A2-2). The underlying idea is that
the relatively greater instability of the augmented chord is a consequence of
the perceptual clarity of the “intervallic equidistance” relative to that heard in
the diminished chord or the tone clusters. Despite the fact that the intervals
of the augmented chord are large and therefore individually consonant, the
three-tone tension effect is perceptually strong. In contrast, the small interval
dissonance in the diminished chord or tone clusters may mask the three-tone
tension effect. Further experimental work is needed here.
In effect, the normalization procedure reduces all 3-tone combinations to
three classes of chord (resolved, tension and dissonant). Calculation of the total
instability of any chord will then result in: (i) a large dissonance factor for the
chords containing dissonant intervals (regardless of the stability/instability of
the chordal factor), (ii) a large inharmoniousness factor for the tension chords
(regardless of the consonance/dissonance of the interval factor), and (iii) low
dissonance and low inharmoniousness factors for the resolved major and minor
chords. The only exceptions are the diminished chords in inverted positions
when only the fundamental frequency is considered (Table A2-1, Column
B1). When upper partials are included in the calculation, however, the
tension factor among various combinations of F0, F1, etc. tones introduces a
significant tension component that results in large instability values for these
chords as well.
As to the relative sonority of the various chord inversions, the results from
non-musicians (Figure 3-11a) indicate clear distinctions, whereas those from
musicians (Figure 3-11b) do not. Since it is likely that the musicians in such
an experiment gave responses in accordance with their identification of the
major/minor mode of the triad, rather than a naïve sonority rating, it may be
Table A2-2. The total “instability” of several types of chords, as calculated using
Eq. A2-2 and three upper partials (F0∼F2). Note that a clear numerical distinction
between the resolved and unresolved chords is obtained.
Chord Type Total instability
C -E-G# Augmented 3.96
C-C#-G Semitone dissonant 2.97
C-D#-F# Diminished 2.49
C-D-G Whole-tone dissonant 1.49
C-D#-G Minor 0.45
C-E-G Major 0.41