DISSONANCE: CRITICAL BAND

James Tenney, in History of Consonance and Dissonance, and William Sethares, in Tuning, Timbre, Spectrum, Scale (p.77), warn us that dissonance cannot be reduced to a single phenomenon or formula.

Nevertheless, the beatings described by Hermann von Helmholtz give us a concrete clue about the origin of the discomfort we perceive when two sounds seem to conflict. Helmholtz identified the key point: when two sound waves are very close, they begin to interfere. They are no longer perceived as separate entities, but as an unstable structure marked by fluctuations. This instability is one of the main physical causes of dissonance. The region where frequencies are close enough to interact and fuse is called the critical band. Within it, beating and irregularities emerge, making perception less stable. Outside it, sounds tend to separate and become more distinct.

CRITICAL BANDS

The original model, however, was not sufficiently detailed. In the following years, Eberhard Zwicker introduced the Bark scale (building on the work of Heinrich Barkhausen), showing that the width of the critical band varies across the frequency range. At low frequencies, a critical band spans only a few hertz but many semitones; at high frequencies, it spans many hertz but few semitones. Later, studies by Brian C. J. Moore and Brian R. Glasberg refined this view, demonstrating that critical bandwidth changes continuously with frequency. For this reason, we do not have a single critical band, but a set of bands distributed across the spectrum, reflecting more accurately how we perceive sound.

Today, critical bandwidth is commonly estimated using the ERB (Equivalent Rectangular Bandwidth).

ERB = 24.7 · (4.37 · f/1000 + 1)

where f is the center frequency in Hz.

EXAMPLE

If we take a C at 130.8 Hz, the ERB is about 38.8 Hz, corresponding to roughly 4.6 semitones.

DISSONANCE

Being inside the critical band does not automatically imply dissonance. Frequencies within this range interact strongly, increasing the likelihood of beating between partials.

According to Reinier Plomp and Willem Levelt, maximum dissonance occurs at approximately 25–35% of the critical band.

This defines three regions:

• beginning of the band → sounds still fused, increasing instability

• circa 30% of the band → maximum dissonance

• beyond 30% → progressive separation and stabilization

Nota	Freq (Hz)	ERB (Hz)	0.3 ERB (Hz)	ERB (st)	0.3 ERB (st)
C1	32.7	28.2	8.5	11.4	3.9
C2	65.4	31.8	9.5	7.0	2.4
C3	130.8	38.8	11.6	4.6	1.5
C4 MIDDLE C	261.6	52.9	15.9	3.3	1.0
C5	523.3	80.1	24.0	2.4	0.8
C6	1046.5	136.6	41.0	2.1	0.7
C7	2093.0	252.0	75.6	2.0	0.6
C8	4186.0	505.0	151.5	2.0	0.6

Critical bands and maximum dissonance points

CONCLUSION

In the low register, even relatively wide intervals can result in instability; in the high register, narrower intervals can be handled with greater control. This also applies within timbre: the sinusoids (partials) that make up a sound interact according to the same principles. Dissonance does not depend only on the distance between fundamentals, but on the interaction between partials within the critical band.

REFERENCES

• The Science of Sound – Thomas D. Rossing (pp. 88, 166)

• Acoustics and Psychoacoustics – David M. Howard & Jamie A. S. Angus (pp. 86, 156)

• History of Consonance and Dissonance – James Tenney

• Tuning, Timbre, Spectrum, Scale – William Sethares