Acoustic pi: Difference between revisions
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Distinguish from logarithmic pi; note its lack of psychoacoustic significance; +category |
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| Line 2: | Line 2: | ||
| Ratio = \pi | | Ratio = \pi | ||
| Cents = 1981.7953553667824 | | Cents = 1981.7953553667824 | ||
| Name = | | Name = acoustic pi | ||
}} | }} | ||
{{Wikipedia|Pi}} | {{Wikipedia|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 [[cent]]s. It is unclear what psychoacoustic significance this interval might have. | ||
Intervals that are close to it are [[3/1]], [[22/7]], and [[355/113]]. | Intervals that are close to it are [[3/1]], [[22/7]], and [[355/113]]. | ||
== Equal divisions == | == Equal divisions == | ||
Using 3. | Using 3.14159…/1 as an interval of equivalence (known as the "pitave") results in some interesting nonoctave tunings. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+Selected edπ–edo correspondence | ||
!N | ! ''N'' | ||
!Description | ! Description | ||
|- | |- | ||
|[[2edπ]] | | [[2edπ]] | ||
|A stack of two minor sevenths, represents a problem of squaring the circle | | A stack of two minor sevenths, represents a problem of squaring the circle | ||
|- | |- | ||
|[[3edπ]] | | [[3edπ]] | ||
|A stack of three compressed fifths, vaguely equivalent to [[2edo]] | | A stack of three compressed fifths, vaguely equivalent to [[2edo]] | ||
|- | |- | ||
|[[4edπ]] | | [[4edπ]] | ||
|Close to equal multiplication of 4/3 | | Close to equal multiplication of 4/3 | ||
|- | |- | ||
|[[5edπ]] | | [[5edπ]] | ||
|Close to equal multiplication of 5/4, [[3edo]] | | Close to equal multiplication of 5/4, [[3edo]] | ||
|- | |- | ||
|[[6edπ]] | | [[6edπ]] | ||
|Close to equal multiplication of 6/5, [[4edo]] | | Close to equal multiplication of 6/5, [[4edo]] | ||
|- | |- | ||
|[[20edπ]] | | [[20edπ]] | ||
|Close to [[12edo]]. | | Close to [[12edo]]. | ||
|- | |- | ||
[[30edπ]] | | [[30edπ]] | ||
|Close to [[18edo]], but sets fractional temperaments to 4:5:6 triad. | | Close to [[18edo]], but sets fractional temperaments to 4:5:6 triad. | ||
|- | |- | ||
|[[38edπ]] | | [[38edπ]] | ||
|Very close to [[23edo]] | | Very close to [[23edo]] | ||
|- | |- | ||
|71edπ | | 71edπ | ||
|Very close to [[43edo]] | | Very close to [[43edo]] | ||
|- | |- | ||
|109edπ | | 109edπ | ||
|Extremely close to [[66edo]] | | Extremely close to [[66edo]] | ||
|} | |} | ||
| Line 52: | Line 53: | ||
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. | 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. | ||
[[Category:Transcendental]] | |||
Revision as of 08:31, 2 April 2023
| Interval information |
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.