Acoustic pi

This page presents a topic of primarily mathematical interest. While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown.

Interval information

Expression	[math]\displaystyle{ \pi }[/math]
Size in cents	1981.795¢
Name	acoustic pi

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Pi

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. Octave-equivalent intervals include acoustic tau (3181.795 cents) and reduced acoustic pi (781.795 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.

See also

• Lucy tuning
• Pi-edo
• Radian

External links

• pi, π on Tonalsoft Encyclopedia