From Note to Hertz: How Equal Temperament Turns Pitch into a Number

Hit the A above middle C on a well-tuned piano and the string vibrates 440 times a second — 440 Hz. Move up one key and the frequency rises, but not by a fixed number of hertz: it rises by a fixed ratio. That distinction is the whole secret of modern tuning. Once you know the single ratio that separates one semitone from the next, you can calculate the exact frequency of any note in any octave from a single reference pitch, and understand why pianos, synths and tuners all agree on what “an A” is.

A semitone is a ratio, not a step

Western music divides the octave into twelve equal semitones, and an octave is a doubling of frequency — A4 is 440 Hz, A5 is 880 Hz, A3 is 220 Hz. If twelve equal steps have to multiply up to exactly double, then each step must multiply the frequency by the twelfth root of two:

2^(1/12) ≈ 1.059463

That’s the engine of equal temperament. Every semitone you go up multiplies the frequency by about 1.0595; every semitone down divides by it. Twelve of those multiplications in a row land you on exactly 2× — a perfect octave — which is the property the whole system is built to guarantee. It’s why no key sounds more “in tune” than any other on a piano: the steps are mathematically identical everywhere on the keyboard.

One formula for every note

To turn that ratio into actual frequencies you need a reference note and a way to count how many semitones away your target is. The convention is to count in MIDI note numbers, where A4 is number 69, and to anchor the whole grid to the frequency of A4:

f = A4 × 2^((MIDI − 69) / 12)

Plug in MIDI 69 and the exponent is zero, so f = A4 = 440 Hz, as it must. Middle C is MIDI 60 — nine semitones below A4 — so its frequency is 440 × 2^(−9/12) ≈ 261.63 Hz. Go an octave up to C5 (MIDI 72) and you get 523.25 Hz, exactly double, confirming the octave relationship falls out of the same equation. This is precisely what the Musical Note Frequency Calculator computes: pick a note and octave, and it counts the semitones from A4 and raises two to the appropriate power.

Why A4 = 440 isn’t the only choice

The formula needs an anchor, and 440 Hz is the modern international standard for A4 — but it isn’t sacred. Change the reference and every note shifts with it, because the whole grid hangs off that one number. Some ensembles tune to 442 or 444 Hz for a slightly brighter, more projecting sound; many orchestras have crept upward over the centuries. And 432 Hz has a devoted following who find it warmer, even though there’s no acoustic magic to it — it’s simply a different anchor that drags the entire scale down by about 32 cents. The calculator lets you set A4 to any of these, and you’ll see every frequency in the table move in lockstep, which is the clearest demonstration that pitch standards are a human agreement, not a law of physics.

Cents: measuring the cracks between the keys

Twelve semitones per octave is fine for notes, but tuning lives in the gaps between them. A cent is one hundredth of a semitone, so there are 1,200 cents in an octave, and a shift of c cents multiplies the frequency by:

2^(c / 1200)

Cents are how you describe being slightly sharp or flat without leaving the note. A guitar string that’s “10 cents flat” is a tenth of the way down toward the note below — audible to a careful ear, and exactly the resolution a tuner displays. Because cents are logarithmic like everything else here, the same number of cents represents the same musical distance whether you’re tuning a low bass note or a high piccolo, even though the raw hertz difference is far larger up high.

Frequency, period and where it meets the studio

Every frequency has a flip side: its period, the time for one cycle, which is just one divided by the frequency. A4 at 440 Hz repeats every 1/440 second ≈ 2.27 milliseconds. That number is the bridge from music theory to audio engineering — it’s why the period readout sits right next to the frequency in the tool. Knowing the period of a note matters when you’re aligning oscillators, setting up a sync, or thinking about how a waveform interacts with a delay line.

When you want the exact hertz for any note — to tune a synth oscillator, check an instrument against a reference, or transpose a part — open the Musical Note Frequency Calculator, choose your note, octave and A4 reference, and nudge the cents slider to model a detune. If your work is more about timing than pitch — syncing delays and LFOs to a track — pair it with the BPM to Milliseconds Delay Calculator, and you’ll have both halves of the frequency-and-time picture covered. Learn the twelfth-root-of-two once and every note on the keyboard becomes a number you can predict.