In the well-known 1965 paper 'Tonal Consonance and Critical Bandwidth', Reinier Plomp and Willem Levelt used individual sine wave pairs in order to investigate consonance and dissonance.  The work formed a consistently credited view of what consonance and dissonance are, taking up its place on a respectable continuum:  
1. Simple integer frequency ratios (Pythagoras)
2. Harmonic Relationships (Rameau)
3. Beats between upper harmonics (Helmholtz)
4. Difference Tones (Krueger)
5. Perceptual Fusion (Stumpf)
[As usual, mouseover any green text to display more information]

"The impact of the critical band observation was to give a plausible physical mechanism for the perception of tonal dissonance. So rather than saying it's a "perception" that is fully in the mind, this allows one to have a mechanism for how the perception comes about."
- William Sethares

Fig. 1 - Interval Consonance vs. Critical Band
The Experiment
Musically untrained subjects were chosen as listeners, and this has proved rather controversial. I'd like to expand upon this in a future entry, but suffice to currently state my agreement based on music's larger lay-audience as opposed to trained ears with preference towards common intervals. Subjects were presented with sine wave interval pairs and asked to judge them on a scale as consonant or dissonant. Tone pairs were selected based only upon common mean frequencies, and no bias was held  towards existing 'known' intervals. 

The Results
The interim results produced by the experiment can be seen in Figure 1, and show us that maximum dissonance can be seen when two sine wave pairs span 25% of the critical band, while increasing deviation from this leads to increasing consonance. The findings were extrapolated by addition into results for complex tones, resulting in the graph below. Theoretical results of the consonance of complex tones based on that of pure tones again pointed towards strong consonance when separated by simple integer frequency ratios. This backed up many past consonance explanations, but for a new, physiological reason.
Fig. 2 - Dissonance vs Two Theoretical Complex Tones
Or did it? A closer look at the published graph reveals: yes, unity as the most consonant, followed by the octave and the fifth, but the next highest is actually the major sixth (3:5).

It's not a world-stopping revelation, indeed the graph changes as different arrangements of harmonics are added to or subtracted from our theoretical complex tones, which can skew the graph in favour of other intervals. This view of consonance priorities and the major sixth is in line with the subjective test of (Malmberg, 1918), but isn't within the traditionally put (or assumed) rank of simplest numbers at the top. Even Pythagoras seems to have missed the major sixth, using the tetractys to scoop the fourth instead.

People practicing n-limit Just Intonation tuning should be aware of what this could imply: The smallest integers don't necessarily produce the most consonance, and harmonic make-up can have a large effect on what sounds best in different situations. There's always room to think with your ears.
9/7/2012 08:42:46

Hmm... What exact math was used to produce figure 2? The octave and unison obviously aren't the same, yet they are measured as having the same consonance.

9/7/2012 08:51:54

Nice observation. Figure 2 was completed using one complex tone of fundamental 250Hz with six harmonics, and another tone varying only in frequency (up to 500Hz) Although it's not 'raw' data from listening tests, it's an elaboration of the results of Figure 1, taking the critical bandwidth across the frequency range and the interaction of all 12 harmonics into account.

Plomp and Levelt point out that it was speculation as to whether one dissonance can simply be added to another in order to arrive at a 'total dissonance', but that's exactly what the graph shows - the theoretical total dissonance if the dissonance of all the interactions were added together. It should be noted that total dissonance is not necessarily the inverse of total consonance. It was a very theoretical elaboration, and a first attempt at exploring the links of critical bandwidth and consonance/dissonance.

While the octave and unison seem to reach the same peak, (perhaps with totally harmonic, synthesised tones this can be a fair claim), the unison has a sharper peak than the octave. This means our tolerance towards a mistuned octave is bigger than that towards a mistuned unison, so there is some difference there. The real world will, of course, be less forgiving than a lab experiment.


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    Jim Russell

    Research and Writings on Musical Tuning and Temperament