E2 note frequency calculator
E2 is MIDI note 40, -29 semitones from A4. At the standard A4 = 440 Hz that puts it at 82.407 Hz.
- Period
- 0.012 134 9 s
- Wavelength in air
- 4.1623
- Quarter wavelength
- 1.0406
- One octave up
- 164.8138 Hz
- One octave down
- 41.2034 Hz
- Equal-tempered fifth above
- 123.4708 Hz
- Just fifth above
- 123.6103 Hz
- How flat the tempered fifth is
- 1.9550 cents
- String length at that tension
- 0.7675
- MIDI note number
- 40
Show the working
- Reference pitch for A4 = 440
- Detune = 0
- Speed of sound = 343
- String tension = 80
- String mass per metre = 0.005
- semitones = -29 = -29
- Frequency = 440 * 2^(-29/12) * 2^(0/1200) = 82.4069
- Period = 82.4069 > 0 ? 1/82.4069 : 0 = 0.012 134 9
- Wavelength in air = 82.4069 > 0 ? 343 / 82.4069 : 0 = 4.162 27
- Quarter wavelength = 4.162 27 / 4 = 1.040 57
- One octave up = 82.4069 * 2 = 164.814
- One octave down = 82.4069 / 2 = 41.2034
- Equal-tempered fifth above = 82.4069 * 2^(7/12) = 123.471
- Just fifth above = 82.4069 * 1.5 = 123.61
- How flat the tempered fifth is = 123.471 > 0 ? 1200 * log2(123.61/123.471) : 0 = 1.955
- String length at that tension = 82.4069 > 0 ? sqrt(80 / 0.005) / (2 * 82.4069) : 0 = 0.767 479
- MIDI note number = 40 = 40
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- f = 440 x 2^((n - 69)/12), with n the MIDI note number
- one cent is 2^(1/1200), so a detune of c cents multiplies by 2^(c/1200)
- string frequency = sqrt(tension / linear density) / (2 x length)
Equal temperament divides the octave into twelve equal ratios of 2^(1/12), which is irrational. Every interval except the octave is therefore slightly out of tune against the simple whole-number ratios the ear prefers.
The tempered fifth is about 1.955 cents flat of a just fifth, which is close enough that almost nobody notices.
Concert pitch is not universal: many orchestras tune to 442 or 443 Hz, and baroque performance often uses 415.
E2 at different concert pitches
| A4 reference | E2 frequency |
|---|---|
| 415 Hz | 77.725 Hz |
| 430 Hz | 80.534 Hz |
| 432 Hz | 80.909 Hz |
| 435 Hz | 81.470 Hz |
| 440 Hz | 82.407 Hz |
| 442 Hz | 82.781 Hz |
| 443 Hz | 82.969 Hz |
| 444 Hz | 83.156 Hz |
Frequently asked
What frequency is E2?
82.407 Hz in equal temperament with A4 at 440 Hz.
How is the frequency of a note calculated?
Multiply the reference pitch by two raised to the number of semitones divided by twelve.