E3 note frequency calculator
E3 is MIDI note 52, -17 semitones from A4. At the standard A4 = 440 Hz that puts it at 164.814 Hz.
- Period
- 0.006 067 45 s
- Wavelength in air
- 2.0811
- Quarter wavelength
- 0.5203
- One octave up
- 329.6276 Hz
- One octave down
- 82.4069 Hz
- Equal-tempered fifth above
- 246.9417 Hz
- Just fifth above
- 247.2207 Hz
- How flat the tempered fifth is
- 1.9550 cents
- String length at that tension
- 0.3837
- MIDI note number
- 52
Show the working
- Reference pitch for A4 = 440
- Detune = 0
- Speed of sound = 343
- String tension = 80
- String mass per metre = 0.005
- semitones = -17 = -17
- Frequency = 440 * 2^(-17/12) * 2^(0/1200) = 164.814
- Period = 164.814 > 0 ? 1/164.814 : 0 = 0.006 067 45
- Wavelength in air = 164.814 > 0 ? 343 / 164.814 : 0 = 2.081 14
- Quarter wavelength = 2.081 14 / 4 = 0.520 284
- One octave up = 164.814 * 2 = 329.628
- One octave down = 164.814 / 2 = 82.4069
- Equal-tempered fifth above = 164.814 * 2^(7/12) = 246.942
- Just fifth above = 164.814 * 1.5 = 247.221
- How flat the tempered fifth is = 246.942 > 0 ? 1200 * log2(247.221/246.942) : 0 = 1.955
- String length at that tension = 164.814 > 0 ? sqrt(80 / 0.005) / (2 * 164.814) : 0 = 0.383 739
- MIDI note number = 52 = 52
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.
E3 at different concert pitches
| A4 reference | E3 frequency |
|---|---|
| 415 Hz | 155.449 Hz |
| 430 Hz | 161.068 Hz |
| 432 Hz | 161.817 Hz |
| 435 Hz | 162.941 Hz |
| 440 Hz | 164.814 Hz |
| 442 Hz | 165.563 Hz |
| 443 Hz | 165.938 Hz |
| 444 Hz | 166.312 Hz |
Frequently asked
What frequency is E3?
164.814 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.