Science & Units
Musical Note Frequency Table
Every frequency in this table comes from one number and one rule. The number is A4 = 440 Hz, fixed by ISO 16 in 1975; the rule is that each semitone multiplies frequency by the twelfth root of two, about 1.059463. Multiply or divide twelve times and you have an octave, which doubles or halves the frequency exactly. That is the whole system of twelve-tone equal temperament, and it is why a piano can be tuned from a single reference.
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The rule is a compromise, and the table makes the compromise visible. A perfect fifth is a 3:2 frequency ratio, which is 701.955 cents; equal temperament makes it exactly 700 cents. A major third is 5:4, or 386.314 cents, against 400 in equal temperament — a gap of nearly 14 cents, which is wide enough to hear as a slight beating when a piano plays a major third against a sustained tone. In exchange, every key sounds equally in tune, which is the property that made the system win.
Two practical notes about the numbers. First, octave numbering is not universal: scientific pitch notation puts middle C at C4, but MIDI puts it at C3 (note 60), so a MIDI reference will disagree with this table by one octave while agreeing on every frequency. Second, the tuning standard is not fixed at 440 — baroque ensembles use 415, some brass bands 442, and the whole table shifts by the same ratio when you change the reference.
Frequency in hertz for every note, octaves 1 to 7
| Octave | C | C♯ | D | D♯ | E | F | F♯ | G | G♯ | A | A♯ | B |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 32.70 | 34.65 | 36.71 | 38.89 | 41.20 | 43.65 | 46.25 | 49.00 | 51.91 | 55.00 | 58.27 | 61.74 |
| 2 | 65.41 | 69.30 | 73.42 | 77.78 | 82.41 | 87.31 | 92.50 | 98.00 | 103.83 | 110.00 | 116.54 | 123.47 |
| 3 | 130.81 | 138.59 | 146.83 | 155.56 | 164.81 | 174.61 | 185.00 | 196.00 | 207.65 | 220.00 | 233.08 | 246.94 |
| 4 | 261.63 | 277.18 | 293.66 | 311.13 | 329.63 | 349.23 | 369.99 | 392.00 | 415.30 | 440.00 | 466.16 | 493.88 |
| 5 | 523.25 | 554.37 | 587.33 | 622.25 | 659.26 | 698.46 | 739.99 | 783.99 | 830.61 | 880.00 | 932.33 | 987.77 |
| 6 | 1,046.50 | 1,108.73 | 1,174.66 | 1,244.51 | 1,318.51 | 1,396.91 | 1,479.98 | 1,567.98 | 1,661.22 | 1,760.00 | 1,864.66 | 1,975.53 |
| 7 | 2,093.00 | 2,217.46 | 2,349.32 | 2,489.02 | 2,637.02 | 2,793.83 | 2,959.96 | 3,135.96 | 3,322.44 | 3,520.00 | 3,729.31 | 3,951.07 |
Each row is one octave: every frequency doubles from the row above. The C in octave 4 is middle C at 261.63 Hz, and A4 is 440 Hz by definition — everything else in the table follows from those two facts.
Interval, frequency ratio and size in cents
| Interval | Semitones | Just ratio | Equal-tempered ratio | Cents |
|---|---|---|---|---|
| Unison | 0 | 1:1 | 1.000000 | 0.0 |
| Minor second | 1 | 16:15 (just) | 1.059463 | 100.0 |
| Major second | 2 | 9:8 (just) | 1.122462 | 200.0 |
| Minor third | 3 | 6:5 (just) | 1.189207 | 300.0 |
| Major third | 4 | 5:4 (just) | 1.259921 | 400.0 |
| Perfect fourth | 5 | 4:3 (just) | 1.334840 | 500.0 |
| Tritone | 6 | 45:32 (just) | 1.414214 | 600.0 |
| Perfect fifth | 7 | 3:2 (just) | 1.498307 | 700.0 |
| Minor sixth | 8 | 8:5 (just) | 1.587401 | 800.0 |
| Major sixth | 9 | 5:3 (just) | 1.681793 | 900.0 |
| Minor seventh | 10 | 9:5 (just) | 1.781797 | 1,000.0 |
| Major seventh | 11 | 15:8 (just) | 1.887749 | 1,100.0 |
| Octave | 12 | 2:1 | 2.000000 | 1,200.0 |
Equal temperament makes every semitone exactly 100 cents, so a perfect fifth is 700 cents instead of the 701.955 cents of the 3:2 ratio. That 2-cent gap is why a piano's fifths beat slightly — it is the price of being able to change key without retuning.
Common tuning standards
| Standard | A4 (Hz) | C4 (Hz) | Where you meet it |
|---|---|---|---|
| ISO 16 / modern concert pitch | 440 | 261.63 | Orchestras, pianos, tuners, most recordings |
| Baroque pitch | 415 | 246.79 | Period-instrument performances of pre-1800 music |
| Classical pitch | 430 | 255.70 | Some historically informed ensembles |
| Verdi / French pitch | 432 | 256.87 | Claimed to be 'natural'; no acoustic basis for the claim |
| Brass band / brass instruments | 442 | 262.81 | Brass bands and some European orchestras |
The C4 column is derived from A4 with the same 2^(−9/12) factor as the main table, so the two always agree. Changing A4 shifts every note by the same ratio, not by a fixed number of hertz.
How to Use These Tables
- To find any note's frequency, find the octave row and the note column. Frequencies below 100 Hz are shown to two decimals because a tenth of a hertz matters more down there: at C1 the whole semitone is only about 2 Hz wide.
- To retune a table to a different standard, multiply every value by newA4 ÷ 440. For baroque pitch at 415 that factor is 0.94318, so C4 moves from 261.63 Hz to 246.79 Hz.
- To check whether an interval will beat, subtract the equal-tempered cents from the just cents: a major third differs by 13.7 cents, a perfect fifth by 2 cents, and an octave by zero. Larger gaps beat faster and are easier to hear.
- To set a tuner or a synth, use A4 as the reference and derive the rest by semitone count. Entering 440 for A4 and using equal temperament will reproduce every value in the first table exactly.
Common Questions
Why is A4 440 Hz and not something round like 400?
It is a convention that hardened rather than a physical constant. A 440 Hz standard was recommended by an international conference in 1939 and formalised by ISO 16 in 1975, largely because American and British instrument makers had already settled there. Before that, A ranged roughly from 415 to 455 depending on the city, which is why period-instrument groups still tune to 415.
Is 432 Hz tuning actually better?
No measurable acoustic property makes 432 Hz superior. It is 31.8 cents flat of 440, which is about a third of a semitone, and it changes nothing about the ratios between notes — the intervals are identical, only the reference moves. Claims about it aligning with nature or the human body do not survive measurement.
What frequency is middle C?
261.63 Hz in scientific pitch notation, where middle C is C4. In MIDI's numbering the same key is called C3, so a MIDI chart will label it one octave lower while giving the same frequency. The number 261.63 assumes A4 = 440; at baroque pitch it is 246.79 Hz.
How far apart are two adjacent notes?
A factor of 2^(1/12), which is 1.059463 — about 5.9%. That is constant across the keyboard, which is the defining feature of equal temperament. In hertz the gap grows with pitch: about 1.9 Hz between C1 and C♯1, and about 30 Hz between C7 and C♯7.
Why does a piano's perfect fifth sound slightly out of tune?
Because equal temperament makes it 700 cents while the pure 3:2 ratio is 701.955 cents. The 2-cent error is small but cumulative: stack twelve fifths and you are 23.5 cents sharp of seven octaves, which is the Pythagorean comma. Tuning a piano spreads that comma across the keyboard instead of leaving it in one place.
More Science & Units
Sources
- ISO 16:1975 — Acoustics: standard tuning frequency — International Organization for Standardization
- Equal temperament and musical intervals — HyperPhysics, Georgia State University
- The Science of Musical Sound — National Institute of Standards and Technology
- Measurement of musical pitch and the history of A4 — The Journal of the Acoustical Society of America
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