Maximal dissonance tuning

A maximal dissonance tuning is a tuning where the intervals between consecutive pitches are selected to maximize sensory dissonance according to a psychoacoustic model of the human inner ear.

There are several published formulas on which a maximal dissonance tuning can be based. One of the earliest (not necessarily the most accurate) was found by Kameoka & Kuriyagawa in 1969, who played two sine waves simultaneously at equal intensity for human listeners. They computed based on experiments that if the lower sine wave has frequency f Hz, then the upper frequency that maximizes sensory dissonance is about [math]\displaystyle{ D(f) = f + 2.27 f^{0.477}\ \text{Hz} }[/math]. The ratio [math]\displaystyle{ \frac{D(f)}{f} }[/math] changes slowly with absolute frequency. It is about 3.22 semitones at 100 Hz, 1.55 semitones at 440 Hz, and 1.03 semitones at 1000 Hz.

From the Kameoka & Kuriyagawa formula, a maximal dissonance tuning can be constructed. Start with a low frequency such as f = 20 Hz and iteratively compute [math]\displaystyle{ D(f) }[/math], [math]\displaystyle{ D(D(f)) }[/math], etc. Collect all these frequencies and the result is a maximal dissonance tuning. As [math]\displaystyle{ \frac{D(f)}{f} }[/math] is frequency-dependent, the resulting tuning is not periodic and can't even be arbitrarily transposed, but small amounts won't upend the overall psychoacoustic effect.

Other maximal dissonance tunings are possible, using different formulas based on assumptions of more complex timbres than sine waves. Simply iterating a function like [math]\displaystyle{ D(f) }[/math] makes the tuning only psychoacoustically interesting for consecutive scale steps. It may be interesting to, for example, optimize every group of n > 2 consecutive pitches to maximize dissonance.