Acoustic e: Difference between revisions
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'''e''' is a mathematical constant associated with the natural logarithm. Because pitch is logarithmic with respect to frequency, it might be of interest in xenharmony as well, where the name ''natave'' (a portmanteau of "natural" and "octave") is suggested. | {{DISPLAYTITLE:Acoustic ''e''}} | ||
{{Infobox Interval | |||
| Ratio = e | |||
| Cents = 1731.2340490667561 | |||
| Name = natave | |||
}} | |||
'''''e''''' is a mathematical constant associated with the natural logarithm. Because pitch is logarithmic with respect to frequency, it might be of interest in xenharmony as well, where the name '''natave''' (a portmanteau of "natural" and "octave") is suggested for the interval ''e'', a.k.a. ''e''/1. | |||
''e'' lies between the JI intervals [[8/3]] and [[11/4]] and is well-approximated by [[49/18]] and [[87/32]], making it a rather sharp eleventh. Due to the limit definition of ''e'', it also occurs as the limit of the following sequence: 2/1, two 3/2, three 4/3, four 5/4, five 6/5, etc. Edos that provide an increasingly close approximation to it are {{EDOs|2, 5, 7, 9, 34, 43, 52, 61, 131, 192, 253}}. | |||
==Approximations== | |||
{{interval edo approximation| interval = 271801/99990}} | |||
== See also == | == See also == | ||
* [[ | * [[Ede|Ed''e'']], equal divisions of this interval | ||
* [[Zetave]], the result of stacking 𝜏 nataves, of interest in regards to the [[zeta]] function | |||
[[Category:Transcendental]] | [[Category:Transcendental]] | ||
Latest revision as of 23:45, 12 December 2025
| Interval information |
e is a mathematical constant associated with the natural logarithm. Because pitch is logarithmic with respect to frequency, it might be of interest in xenharmony as well, where the name natave (a portmanteau of "natural" and "octave") is suggested for the interval e, a.k.a. e/1.
e lies between the JI intervals 8/3 and 11/4 and is well-approximated by 49/18 and 87/32, making it a rather sharp eleventh. Due to the limit definition of e, it also occurs as the limit of the following sequence: 2/1, two 3/2, three 4/3, four 5/4, five 6/5, etc. Edos that provide an increasingly close approximation to it are 2, 5, 7, 9, 34, 43, 52, 61, 131, 192, 253.
Approximations
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 7 | 10\7 | 1714.29 | -16.95 | -9.89 |
| 9 | 13\9 | 1733.33 | +2.10 | +1.57 |
| 16 | 23\16 | 1725.00 | -6.23 | -8.31 |
| 18 | 26\18 | 1733.33 | +2.10 | +3.15 |
| 25 | 36\25 | 1728.00 | -3.23 | -6.74 |
| 27 | 39\27 | 1733.33 | +2.10 | +4.72 |
| 34 | 49\34 | 1729.41 | -1.82 | -5.16 |
| 36 | 52\36 | 1733.33 | +2.10 | +6.30 |
| 43 | 62\43 | 1730.23 | -1.00 | -3.59 |
| 45 | 65\45 | 1733.33 | +2.10 | +7.87 |
| 52 | 75\52 | 1730.77 | -0.46 | -2.01 |
| 54 | 78\54 | 1733.33 | +2.10 | +9.45 |
| 61 | 88\61 | 1731.15 | -0.09 | -0.44 |
| 70 | 101\70 | 1731.43 | +0.19 | +1.13 |
| 77 | 111\77 | 1729.87 | -1.36 | -8.75 |
| 79 | 114\79 | 1731.65 | +0.41 | +2.71 |