Acoustic pi: Difference between revisions
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{{Novelty}} | |||
{{Infobox Interval | |||
| Ratio = \pi | |||
| Cents = 1981.7953553667824 | |||
| 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. Octave-[[equivalent]] intervals include '''acoustic tau''' (3181.795 [[cent]]s) and '''reduced acoustic pi''' (781.795 cents). It is unclear what psychoacoustic significance these intervals might have. | ||
Intervals that are close to | Intervals that are close to the acoustic pi 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. | ||
== Approximations == | |||
{{interval edo approximation|interval=355/113}} | |||
{| class="wikitable" | |||
|+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 the 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]] | |||
* [[Acoustic phi]] | |||
* [[Acoustic e]] | |||
* [[Edϕ]] | |||
* [[EDe]] | |||
* [[User:Eliora/Phi to the phi]] | |||
== External links == | |||
* [http://tonalsoft.com/enc/p/pi.aspx pi, π] on [[Tonalsoft Encyclopedia]] | |||
[[Category:Transcendental]] | |||
Latest revision as of 00:10, 18 February 2026
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This page presents a novelty topic.
It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks. |
| 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. Octave-equivalent intervals include acoustic tau (3181.795 cents) and reduced acoustic pi (781.795 cents). It is unclear what psychoacoustic significance these intervals might have.
Intervals that are close to the acoustic pi 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.
Approximations
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 3 | 5\3 | 2000.00 | +18.20 | +4.55 |
| 6 | 10\6 | 2000.00 | +18.20 | +9.10 |
| 17 | 28\17 | 1976.47 | -5.32 | -7.54 |
| 20 | 33\20 | 1980.00 | -1.80 | -2.99 |
| 23 | 38\23 | 1982.61 | +0.81 | +1.56 |
| 26 | 43\26 | 1984.62 | +2.82 | +6.11 |
| 40 | 66\40 | 1980.00 | -1.80 | -5.99 |
| 43 | 71\43 | 1981.40 | -0.40 | -1.43 |
| 46 | 76\46 | 1982.61 | +0.81 | +3.12 |
| 49 | 81\49 | 1983.67 | +1.88 | +7.67 |
| 60 | 99\60 | 1980.00 | -1.80 | -8.98 |
| 63 | 104\63 | 1980.95 | -0.84 | -4.43 |
| 66 | 109\66 | 1981.82 | +0.02 | +0.12 |
| 69 | 114\69 | 1982.61 | +0.81 | +4.68 |
| 72 | 119\72 | 1983.33 | +1.54 | +9.23 |
| 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 the 3:4:5 triad with a fractional-octave temperament just as 12edo does with the 4:5:6 triad.