Scarlet & Grey
Ohio State University
School of Music

Consonance and Dissonance - Huron's Numerosity Conjecture

Roberts and Mathews (1984) observed that there is a tendency for some listeners to judge just intervals less pleasant than slightly mistuned intervals. Vos (1986) also noted that some people may rate pure intervals to be "insipid" and therefore prefer intervals slightly tempered as expressing greater "warmth."(p.223) At the same time, Vos has shown that listeners judge as "most pure" those intervals that are precisely just -- with no tempering.

In examining typical outputs from Kameoka and Kuriyagawa's measurement algorithm, we observed two unexpected results.

  1. The following graph shows the dissonance values for 13 intervals spanning P1 to P8 with C4 as the lowest note. The tones are complex tones consisting of the first 10 harmonics, where the first harmonic has a sound pressure level of 60 dB, and each subsequent harmonic is reduced by 6 dB. Notice that, when the intervals are rank ordered by dissonance, the largest difference between any successive pair of intervals is the contrast between the unison (P1) and the octave (P8).

    Dissonance Ratings

  2. Another unexpected result is that nearly every 3-note chord is more dissonant than every 2-note chord. Similarly, nearly every 4-note chord is more dissonant than every 3-note chord.

On reflection, the tonotopic account of Plomp and Levelt (1965) implies that adding spectral components will always result in an increase in sensory dissonance.

Tests of Huron's Conjecture

If this view of consonance and dissonance is correct, then increased tonal fusion ought to result in a decrease in the perceived consonance. This prediction is precisely contrary to traditional view such as those associated with Stumpf.

Suppose we have a perfect fifth between two sine tones. We can predict that if a listener hears these "analytically" as two tones, then the stimulus will be perceived as more consonant than if the two pure tones are heard synthetically as a single tone.

The challenge is to figure out how one might trick listeners to hear the stimulus as either one or two tones. One way to do this might be to precede the stimulus by a brief "warning tone". This brief tone might be sounded 1 second before the onset of the main stimulus. In some trials the frequency of the warning tone would be close to the fundamental of one of the two pure tones forming the dyad. This will tend to promote stream segregation between the two tones due to pitch proximity. In other trials the warning tone could occur at some remote distance from the two tones, and so streaming would tend not to break apart the two tones.

If Huron's conjecture is correct, then stimuli that are preceded by distant warning tones should be judged more consonant than identical stimuli preceded by warning tones that are near. Notice that this hypothesis is precisely the reverse of the Wright-Bregman hypothesis.

There is a second, more direct test of Huron's conjecture. If the theory of sensory dissonance is correct (that dissonance is a negative-valence emotion in response to stimulus-induced degradation of the auditory system), then here is a potentially useful way of measuring dissonance:

Ask listeners to judge whether they hear 1 or 2 tones. Play the following types of stimuli: either (1) two pure tones with different frequency separations and different mean frequencies. As filler stimuli use either (2) single pure tones or (3) single pure tones with various amounts of amplitude modulation.

Huron would predict finding that the maximum confusion in responding 1 or 2 tones would arise for stimuli type (1) when the tones are separated by roughly 1/2 a critical band.

If this prediction is confirmed then it provides excellent support for the idea that dissonance is related to perceptual confusion

Also, if this prediction is confirmed then having listeners judge whether a stimulus contains 1 or 2 tones provides a much more convenient instruction for experiments than telling listeners to judge the unpleasantness, ugliness, euphoniousness, consonance or dissonance. The instructions would be much clearer.