Acoustic pi

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This page presents a novelty topic. It features ideas which are less likely to find practical applications in xenharmonic music. It may contain numbers that are impractically large, exceedingly complex or chosen arbitrarily.

Novelty topics are often developed by a single person or a small group. As such, this page may also feature idiosyncratic terms, notations or conceptual frameworks.

Interval information
Expression [math]\pi[/math]
Size in cents 1981.7954¢
Name acoustic pi
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~4.59121 bits
English Wikipedia has an article on:

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.

Intervals that are close to it are 3/1, 22/7, and 355/113.

Equal divisions

Using 3.14159…/1 as an interval of equivalence (known as the "pitave") results in some interesting nonoctave tunings.

Selected edπ–edo correspondence
N Description
2edπ A stack of two minor sevenths, represents a problem of squaring the circle
3edπ A stack of three compressed fifths, vaguely equivalent to 2edo
4edπ Close to equal multiplication of 4/3
5edπ Close to equal multiplication of 5/4, 3edo
6edπ Close to equal multiplication of 6/5, 4edo
20edπ Close to 12edo.
30edπ Close to 18edo, but sets fractional temperaments to 4:5:6 triad.
38edπ Very close to 23edo
71edπ Very close to 43edo
109edπ 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.