Acoustic pi
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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.
| Interval information |
Equal divisions
Using 3.14159…/1 as an interval of equivalence (known as the "pitave") results in some interesting nonoctave tunings.
| 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.