Cents Calculator: Convert Hz to Cents & Cents to Hz

Last updated: October 23, 2026 · Written by the Pitch Detectors Editorial Team

A cents calculator converts between frequency and cents in both directions: enter two frequencies to get their pitch distance in cents, or one frequency plus a cent offset to get the exact shifted frequency. One cent is 1/1200 of an octave — 100 cents make a semitone, 1200 cents make an octave — on the logarithmic scale the ear actually uses.

Key Takeaways

• 100 cents = one semitone; 1200 cents = one octave. One cent is a frequency ratio of 2^(1/1200) ≈ 1.0005778.
• Two formulas cover every conversion: cents = 1200 × log₂(f₂ ÷ f₁); frequency = f₁ × 2^(cents ÷ 1200).
• Positive cents mean sharp (higher pitch); negative cents mean flat (lower pitch).
• 1 Hz has no fixed cent value: it equals 15.67 cents at 110 Hz but only 1.73 cents at 1000 Hz.
• A440 vs A432 differ by 31.77 cents — one-third of a semitone. Most listeners notice pitch differences of about 5–8 cents between successive tones.

What Is a Musical Cent?

Hertz misleads the ear. A 10 Hz gap between 55 Hz and 65 Hz spans 289.2 cents — nearly three semitones. The same 10 Hz gap between 1760 Hz and 1770 Hz spans 9.8 cents, effectively inaudible. The cent fixes this by measuring ratios instead of raw differences, which matches how hearing works.

The same trap catches anyone converting a tuner reading by hand: 1 Hz equals 15.67 cents at 110 Hz, 3.93 cents at 440 Hz, and 1.73 cents at 1000 Hz. No fixed Hz-per-cent rule exists — always convert with the ratio formula, or let the calculator do it.

How This Calculator Works

The calculator runs in two modes. Frequency to Cents takes two frequencies and returns their distance in cents, the frequency ratio, the same distance in semitones, a plain-language perception label, and the direction — which of the two frequencies is sharper. Cents to Frequency takes a base frequency plus a signed cent offset and returns the exact target frequency. Results update as you type; both modes require positive frequencies, because zero and negative values have no pitch.

Methodology: frequency mode computes c = 1200 × log₂(f₂/f₁); cents mode computes f₂ = f₁ × 2^(c/1200). The formula was checked against the published cent values of 12 just-intonation intervals — the octave (2:1) at exactly 1200, the fifth (3:2) at 701.96, the major third (5:4) at 386.31 — and matched to two decimal places in every case. Last reviewed: October 2026.

The Cents Formulas and Worked Examples

Two formulas cover every conversion this page performs:

Worked example 1 — 440 Hz vs 432 Hz:

1. Form the ratio: 432 ÷ 440 = 0.98182
2. Take the base-2 log: log₂(0.98182) = −0.02647
3. Multiply by 1200: −0.02647 × 1200 = −31.77 cents
4. Read it: 432 Hz sits 31.77 cents flat of 440 Hz — about one-third of a semitone.

Worked example 2 — adding a quarter tone to A440: f₂ = 440 × 2^(50/1200) = 440 × 1.02930 = 452.89 Hz, the exact frequency of a quarter-tone-sharp A.

How to Use This Cents Calculator

1. Choose a mode: "Frequency to Cents" compares two pitches; "Cents to Frequency" applies an offset to one pitch.
2. In Frequency to Cents mode, enter the reference frequency first and the measured frequency second — 440 and 442, for example.
3. In Cents to Frequency mode, enter the base frequency and the offset — positive for sharp, negative for flat, such as 440 and +50.
4. Try a quick button: presets cover 440 vs 432, an octave, semitone steps, +100¢, +1200¢, and the −31.77¢ A432 offset.
5. Read the result panel: the large number is the answer, with the ratio, semitone equivalent, perception label, and direction beneath it.

Reading Your Result

The perception label in the result panel follows fixed bands — and each band calls for a different response:

Direction matters as much as size: sharp means tune down, flat means tune up. String players tune up to the note, because approaching from below keeps string tension stable.

Why 1200? The Story Behind the Number

The number 1200 is a deliberate design choice, not a law of physics. In the 1880s, English mathematician Alexander J. Ellis needed one unit for comparing instrument tunings from around the world. He published the system in his 1885 paper "On the Musical Scales of Various Nations" and in his English edition of Helmholtz's On the Sensations of Tone, building on Gaspard de Prony's logarithmic pitch work from the 1830s.

The arithmetic explains the choice: 12 semitones per octave, and 1200 gives each exactly 100 cents — a decimal grid where musicians think in whole numbers. One cent is the 1200th root of 2 as a frequency ratio (about 1.0005778) — far finer than anyone can hear between successive notes.

The 440 Hz vs 432 Hz Question, Answered in Cents

Some listeners prefer music tuned to A432 instead of the standard A440 (ISO 16, adopted internationally in 1955). In cents, the difference is exactly −31.77 — about one-third of a semitone flat. That gap is plainly audible.

What cents cannot settle is the why. Claims that 432 Hz is "more natural" or carries special healing properties have no support in peer-reviewed acoustics research; controlled listening tests attribute reported preferences to expectation and lowered string tension. The cent value is objective; the interpretation is personal. To experiment, apply −31.77 cents to any A440 frequency in Cents to Frequency mode.

Just Intonation vs Equal Temperament, in Cents

A just interval uses a simple whole-number ratio; the piano's equal-tempered version lands a few cents away from it. That gap — the temperament error — is what a cents calculator exists to measure. The table gives every interval inside the octave from its 5-limit just ratio:

Two patterns stand out. Fifths and fourths sit within 2 cents of their just values (+1.96 and −1.96 cents), which is why equal temperament works at all. Thirds and sixths carry the real cost — each sits 13.69 to 15.64 cents from just — and the minor seventh is widest at +17.60 cents. String quartets and barbershop singers bend toward the just column by ear; fixed-pitch instruments cannot.

Stretch Tuning: Why Pianos Are Never at 0 Cents

A well-tuned piano is deliberately out of tune with the cent ruler. Real strings are stiff, and stiffness makes a string's overtones ring slightly sharper than pure theory predicts — a phenomenon called inharmonicity. Set every note to exactly 0 cents against an electronic reference and the octaves sound dull and beating.

Professional piano tuners therefore use stretch tuning: treble notes are tuned a few cents sharp and bass notes a few cents flat, following the instrument's inharmonicity curve (the Railsback curve). Concert grands can stretch 20–30 cents at the extreme ends. So "0 cents" is the right target for a guitar tuner app — and the wrong target for a Steinway's top octave.

From Cents to Beats: Pitch Math You Can Hear

Two nearly identical pitches sounding together produce beating — a pulsing as the waves drift in and out of phase — and the beat rate equals the plain frequency difference: beats/sec = |f₂ − f₁|.

Example: two A strings at 440 Hz and 2 cents sharp. The sharp one is 440 × 2^(2/1200) = 440.51 Hz, so you hear about 0.5 beats per second — one gentle pulse every two seconds. Tuners use exactly this: slow the beats to zero and the unison is perfect. Simultaneous tones reveal 1–2 cent differences that pass unnoticed in a melody.

Cents to Percent: Reading Synth Fine-Tune Knobs

Synth oscillators and DAW detune knobs often display percent instead of cents: percent = (2^(cents ÷ 1200) − 1) × 100. One cent equals about 0.0578%, so the relationship stays nearly linear for small offsets and bends away past ±50 cents.

Under ±25 cents, multiplying cents by 0.0578 lands within a hair of the true value — the shortcut for when a knob speaks percent and the tuner speaks cents.

Limitations

• Cents assume 12-tone equal temperament as the ruler. They measure deviations precisely, but a just-intonation or maqam interval still needs its own ratio as reference.
• Hearing is not a meter. Loudness, timbre, and register shift perceived pitch slightly — identical cent values can feel different in context.
• Extreme registers widen the just-noticeable difference. Below roughly 100 Hz and above roughly 5 kHz, most listeners need larger cent gaps to notice anything.
• Real instruments drift. Temperature, string age, and breath pressure move pitch by several cents during a performance — chasing 0.0 cents on a live instrument is wasted effort.
• The calculator converts numbers, not sound. It converts the values you type; it does not listen to audio.

Cents vs Other Ways to Measure Pitch

Convert between note names and frequencies with the frequency-to-note converter or its reverse, the note-to-frequency converter. To hear these intervals in practice, try the pitch detector or the online guitar tuner.

Frequently Asked Questions