R. Plomp and J. M. Levelt "Tonal Consonance and Critical Bandwidth" Journal of the Acoustical Society of America, 1965.

Sole Concern:

Why consonance is related to simple frequency ratio. Tonal consonance defined here as the characteristic sensory experience associated with isolated tone pairs with simple frequency ratios.

Previous Explanations of Consonance:

1. Frequency Ratio (Galileo, Leibniz, Euler, et al)
2. Relationship of Harmonics (Rameau, von Helmholtz, Wundt, et al)
3. Beats between Harmonics (Sorge, von Helmholtz)
4. Difference Tones (Preyer, Krueger, Sandig)
5. Fusion (Stumpf)
6. Culture (Cazden, Lundin)
7. Genetics (Ogden, Moore)

Critique of Explanations 1, 2, 4, 5, and 6:

Against 1 & 2: Data from Guthrie & Morrill and Kaestner show no peak at simple ratios.
Against 4: nonlinear distortion of hearing organ is too small
Against 5: see van de Geer, Levelt and Plomp 1962.
Against 6: opinion that there exists a sensory phenomenon related to simple integral ratios for musically trained and untrained subjects.

Experiments:

Subjects rated pure intervals on a 7-point "consonant-dissonant" scale "Consonant" defined as "beautiful" and "euphonious" Mean frequencies of 125, 250, 500, 1000, and 2000 Hz were used Each subject judged 12-14 intervals with various frequency differences around these means The data from unreliable subjects was excluded

Results:

Small frequency differences received a minimum rating for consonance, while larger differences show a more or less broad maximum Contra von Helmholtz, the frequency difference for maximum dissonance is not independent of mean frequency Rule of thumb: maximum tonal dissonance at 25% of critical bandwidth, maximal tonal consonance reached at 100% of critical bandwidth

Application to Complex-Tone Intervals:

Consonance as a function of critical bandwidth is plotted
Assuming six harmonics, consonance of various intervals above 250 Hz can be calculated:

Dissonances for each pair of adjacent partials are added
Result shows consonance peaks at integral ratios

Analysis of Chords in Music:

Frequency differences of intervals (including partials) above the Cs and Gs in a Bach trio sonata and a Dvorak string quartet are calculated and plotted. These curves follow the curves for critical bandwidth.

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