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. |
| Expression | [math]\pi[/math] |
| Size in cents | 1981.7954¢ |
| Name | acoustic pi |
| Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~4.23555 bits |
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.
| 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.