The acoustic pi, the transcendental number equal to the ratio of a circle's circumference to the diameter, is about 3.14159, a rather minor thirteenth of 1981.795 cents. It is unclear what psychoacoustic significance this interval might have.
Using 3.14159…/1 as an interval of equivalence (known as the "pitave") results in some interesting nonoctave tunings.
|A stack of two minor sevenths, represents a problem of squaring the circle
|A stack of three compressed fifths, vaguely equivalent to 2edo
|Close to equal multiplication of 4/3
|Close to equal multiplication of 5/4, 3edo
|Close to equal multiplication of 6/5, 4edo
|Close to 12edo.
|Close to 18edo, but sets fractional temperaments to 4:5:6 triad.
|Very close to 23edo
|Very close to 43edo
|Extremely close to 66edo
Temperaments of interest
Engineer's temperament, tempering out π/3, the engineer's comma.
20edπ can be used to set 3:4:5 triad with a fractional-octave temperament just as 12edo does with the 4:5:6 triad.