Maximal dissonance tuning
While musical dissonance depends on many complex factors including the listener's conditioning and cultural background, sensory dissonance or roughness is much more consistent among average human listeners (i.e. those not affected by conditions such as congenital amusia). In its simplest form, sensory dissonance happens when two sine tones are played simultaneously at roughly equal intensity, close enough in frequency that the basilar membrane has trouble distinguishing them but far apart enough that beating is not audible. This is purely about the physiology of the inner ear, and not about approximations to familiar intervals such as JI ratios.
In 1969, Kameoka & Kuriyagawa published the results of a study where listeners were asked to rate the roughness of pairs of sine tones. They calculated an approximate formula that, given a sine tone frequency, produces another frequency above it that ostensibly maximizes the sensation of roughness. If each sine tone is played at a comfortable 57 dB SPL and the lower one has frequency f Hz, their analysis states that the upper sine tone of maximal frequency is given by [math]D(f) = f + 2.27 f^{0.477}\ \text{Hz}[/math]. The ratio [math]\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 [math]D(f)[/math] Kameoka & Kuriyagawa designed a somewhat ad hoc formula for measuring the roughness of any two sine waves played at roughly equal intensity. This was one of the earliest studies into quantification of roughness; later research has poked holes in their methodology and produced somewhat different results.
Regardless of the accuracy of [math]D(f)[/math], it can be used to construct a strange tuning that maximizes the sensory dissonance between consecutive tones. Start with a low frequency such as f = 20 Hz and iteratively compute [math]D(f)[/math], [math]D(D(f))[/math], etc. Collect all these frequencies to produce a tuning. As [math]\frac{D(f)}{f}[/math] is frequency-dependent, the resulting tuning is not periodic and can't even be arbitrarily transposed without ruining its nature (although small transpositions won't upend the overall psychoacoustic effect).
More sophisticated approaches are possible for constructing tunings from sensory dissonance power laws. Simply iterating a function like [math]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 roughness. [math]D(f)[/math] also works on the assumption of sine wave tones, and can be expanded to more complex timbres.