Study link: https://www.nature.com/articles/s41467-024-45812-z
Start with figure 3. Non-musicians generally favour pythagorian style harmonic ratios, while trained musicians tend to prefer equal temperament.
Each of the following intervals in the data confirm this:
- The pythagorian major 3rd ratio 5/4 is slightly flat compared to 5 semitones. Non-musicians nailed it, musicians prefer the sharp ET version.
- The range from 10-11 semitones is rated relatively high by non-musicians. This range includes the 7th harmonic, 7/4
- Even the 11th harmonic, taking place between 6(tritone) and 7(perfect fifth) is rated highly by non-musicians relatively to musicians.
- The "9" major 2nd octave, 9/4 JI, aka 14 semitones, is tuned more perfectly by non musicians (9/4) compared to musicians
Later the authors experiment with a artificial gong style instrument (fig 5). However its harmonics are remarkably close to several standard pythagorean ratios. You see 1.52f0 (slightly sharp perfect 5th 3/2); 3.46f0 (slightly flat harmonic 7th 7/4); 3.92f0 (slightly flat double octave 4/1). These harmonics are so close to a guitar or piano or human voice, it can't really even be considered inharmonic at all.
With this instrument, you get, yet again, more pythagorean intervals:
- sharp major 2nd, 9/8 ratio (makes sense; started with a sharp 3/2 and 9/8 is just (3/2)^2 modulo octaves).
- People still correctly tune the major 3rd 5/4 ratio (slightly less than 5 semitones) even without a 5th harmonic. bonkers
- Sharp 5th, which makes sense as the harmonic is 1.52f0 instead of 1.5f0
- Heavily emphasized harmonic 7th interval between 9 and 10 semitones
Later, the authors experiment with various mis-tunings of the pythagorean intervals (Fig 8). The most highly rated intervals are slightly sharp or flat from JI. This is evidence for JI intervals being pleasing--slightly sharp or flat creates a pleasing rhythm of Just harmony. It's like just harmony but better.
In conclusion, essentially every plot in the study points towards pythagorean, mathematically tuned harmony being more pleasing. Even in surprising ways. I could not find a single data point indicating otherwise, except for indoctrination of musicians towards ET leading to things like preferring sharp major 3rds and not recognizing the harmonic 7th interval.