Cents Calculator: Hz to Cents & Pitch Difference

Last updated: October 31, 2026 · Written by the Pitch Detectors Editorial Team

A cents calculator measures the pitch distance between two frequencies in cents — the logarithmic unit where 100 cents make one semitone and 1200 make an octave. Enter any two frequencies, or a frequency plus a cent offset, to get the exact interval, its direction (sharp or flat), and the matching frequency.

Key Takeaways

• 100 cents = one semitone; 1200 cents = one octave. The cent is 1/1200 of an octave.
• Formula: cents = 1200 × log₂(f₂ ÷ f₁). Reverse: f₂ = f₁ × 2^(cents ÷ 1200).
• Positive cents mean sharp (higher pitch); negative cents mean flat (lower pitch).
• Most listeners notice pitch differences around 5–8 cents; trained ears catch ~2 cents when two notes sound together.
• A440 vs A432 differs by 31.77 cents — about one-third of a semitone.

What Is a Musical Cent?

Hertz alone mislead the ear. A 10 Hz gap between 55 Hz and 65 Hz spans a dramatic 287 cents — more than two semitones. That same 10 Hz gap between 1760 Hz and 1770 Hz is only 9.8 cents, effectively inaudible. The cent fixes this by measuring ratios instead of raw differences, matching how hearing actually works.

Musicians, tuner apps, and piano technicians all speak in cents because the unit stays musically meaningful from the lowest organ pipe to the highest piccolo note.

How This Calculator Works

This calculator runs in two directions. Frequency mode takes two frequencies and returns their distance in cents, the frequency ratio, the equivalent in semitones, and a plain-language perception label. Cents mode takes a base frequency plus a cent offset and returns the exact target frequency — the math tuner technicians use when tempering intervals.

Methodology: frequency-to-cents uses c = 1200 × log₂(f₂/f₁), computed via natural logarithms as c ≈ 1731.234 × ln(f₂/f₁). Cents-to-frequency uses f₂ = f₁ × 2^(c/1200). The sign convention is directional: a positive result means the second frequency is sharp (higher); negative means flat (lower). We verified this calculator's output against published just-intonation interval tables for 12 sample intervals (octave, fifth, major third, and others) — every result matched to two decimal places. 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: a quarter tone is 50 cents. f₂ = 440 × 2^(50/1200) = 440 × 1.02930 = 452.89 Hz. That is the exact frequency a quarter-tone-sharp A.

How to Use This Cents Calculator

1. Choose a mode: "Frequencies → Cents" compares two pitches; "Cents → Frequency" applies an offset to one pitch.
2. In frequency mode, enter the reference frequency in the first box and the target frequency in the second (e.g. 440 and 442).
3. In cents mode, enter the base frequency and the cent offset — positive for sharp, negative for flat (e.g. 440 and +50).
4. Try a quick value: the preset buttons jump to common cases like +100¢ (one semitone), +1200¢ (one octave), or the 440→432 Hz gap.
5. Read the result: the big number is the cent difference; below it you get the frequency ratio, the semitone equivalent, a perception label, and the direction.

Reading Your Result

The cent value tells you how far off a pitch is and whether it matters. Use this table to interpret it:

The direction matters as much as the size: sharp means tune down, flat means tune up. String players tune up to the note rather than down to it, 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 1870s–1880s, English mathematician Alexander J. Ellis needed a unit for comparing the tunings of instruments from around the world — work he published in his 1875 English edition of Helmholtz's On the Sensations of Tone and his 1885 paper "On the Musical Scales of Various Nations." Building on Gaspard de Prony's logarithmic pitch work from the 1830s, Ellis divided the octave into 1200 parts.

Why that number: Western music divides the octave into 12 semitones, and 1200 gives each semitone exactly 100 cents — a decimal-friendly grid where musicians can think in whole numbers. One cent is therefore the 1200th root of 2 as a frequency ratio (about 1.0005778), far below what anyone can hear between successive notes, which makes it a perfect fine-measuring ruler.

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 is plainly audible: it is roughly the distance between being in tune and being noticeably off.

What cents cannot settle is the why. Claims that 432 Hz is "more natural" or has special healing properties have no support in peer-reviewed acoustics research; controlled listening tests attribute reported preferences to expectation and to the slightly darker timbre that comes with lower tension. The cent value is objective — the interpretation is personal. If you want to experiment, this calculator's cents mode converts any A440-based frequency to its A432 equivalent: multiply by 2^(−31.77/1200).

Stretch Tuning: Why Pianos Are Never at 0 Cents

Here is something most cents calculators never mention: 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. If a tuner set every note to exactly 0 cents against an electronic reference, octaves would 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 relative to theoretical pitch, 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 tuner app on a guitar — and the wrong target for the top octave of a Steinway.

From Cents to Beats: Pitch Math You Can Hear

When two nearly identical pitches sound at once, you hear beating — a pulsing as the sound waves drift in and out of phase. The beat rate in pulses per second equals the plain frequency difference: beats/sec = |f₂ − f₁|. Cents translate directly into this.

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. Tuner technicians use exactly this: slow the beats to zero and the unison is perfect. This is also why simultaneous tones reveal 1–2 cent differences that would pass unnoticed in a melody.

Limitations

• Cents assume 12-tone equal temperament as the ruler. They measure deviations beautifully, but describing a just-intonation or maqam interval still needs its own ratio reference.
• Hearing is not a meter. Loudness, timbre, and register all shift perceived pitch slightly — two tones at identical cents can feel different in context.
• Extreme registers widen the just-noticeable difference. Below ~100 Hz and above ~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.

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 what these intervals do in practice, try the pitch detector or the online guitar tuner.

Frequently Asked Questions