Note to Frequency Converter

A piano is tuned wrong on purpose

Not approximately right. Deliberately, measurably wrong, in every interval except the octave — and the amounts are known to two decimal places.

The ear judges consonance by simple whole-number ratios. Two notes at exactly 3:2 lock together; at 5:4 they lock together. Equal temperament does not give you those ratios. It divides the octave into twelve identical steps of 2^(1/12), an irrational number, which means no interval but the octave can land on a whole-number ratio at all.

The errors are not spread evenly, and that is the design rather than an accident. The fifth is off by under two cents, which nobody can hear. The thirds and sixths are off by thirteen to sixteen cents, which is between two and three times the roughly six cents a trained listener resolves on a sustained tone — and thirds are what chords are built from.

So a piano chord is audibly not in tune with itself, and always has been. Play a major third slowly against a drone and the beating is there. We are simply used to it, from birth, and a barbershop quartet or a string section tuning by ear will drift toward the pure intervals instead, because voices and strings can.

Where the error comes from

It is arithmetic, and there is no way around it. Stack twelve pure fifths and you should arrive back at the note you started on, seven octaves up. You do not.

Twelve fifths is (3/2)¹², which is 531441 / 4096. Seven octaves is 2⁷, or 128. The ratio between them is 531441 / 524288 — about 1.0136, or 23.46 cents. That surplus is the Pythagorean comma, and it is nearly a quarter of a semitone.

Since 3 and 2 are different primes, no number of pure fifths will ever land exactly on a power of two. The comma is not a measurement error or a limit of craftsmanship. It is a fact about the integers, and every tuning system in history is a decision about where to put it.

• Pythagorean tuning kept every fifth pure and dumped the whole comma into one interval, the “wolf fifth”, which howled. Music simply avoided that key.
• Meantone sacrificed the fifths slightly to make the major thirds pure — the opposite trade to ours, and it sounds glorious in the keys it serves and unusable in the rest.
• Well temperaments distributed the comma unevenly, so every key was playable but each had its own character. This is the world Bach wrote The Well-Tempered Clavier for — well-tempered, note, not equal-tempered.
• Equal temperament splits the comma into twelve identical pieces, taking 1.955 cents off every fifth. Every key becomes identical, and every third pays for it.

Notice the arithmetic closes exactly: a twelfth of 23.46 cents is 1.955 cents, which is precisely how flat each equal-tempered fifth is. Modulation to any key became free, and the price was the thirds.

440 Hz is a convention, not a constant

Ask what frequency an A is and the answer is another question: whose A, and when. The number is a committee decision from 1955, standardised as ISO 16, and adherence to it is patchy.

The baroque figure is the elegant one. At 415 Hz the offset is 101.27 cents — within one and a third cents of exactly a semitone. That is not a coincidence: it was chosen so a modern keyboard player can read a baroque score and simply transpose down a semitone, and harpsichords are often built to shift between the two pitches mechanically.

The spread between an early-music ensemble at 415 and a European orchestra at 443 is 113 cents, more than a whole semitone. The same written note is a different pitch entirely. And the drift has been upward for two centuries, because a brighter, more brilliant sound tends to win — which is also why 444 is roughly the practical ceiling, since string tension and instrument stress become the limit.

The 432 Hz claim deserves a plain answer, because it is the most-searched thing here. It is 31.77 cents below 440, about a third of a semitone. Every proposition attached to it — that it is a natural frequency, that it is mathematically special, that it is what Verdi or the ancients used — either rests on numerology or does not survive checking against the historical record, which shows pitch varying continuously by city and decade. If you prefer how it sounds, that is a real preference and a perfectly good reason. It is not a physical fact, and the calculator will happily work in it.

Cents, and what a listener can hear

Pitch is perceived logarithmically: what the ear hears as “the same distance” is a constant ratio, not a constant number of hertz. So an octave is always a doubling, whether that means 27.5 Hz to 55 Hz at the bottom of a piano or 2093 Hz to 4186 Hz at the top.

This is why hertz are a poor way to talk about being out of tune. Being 2 Hz sharp on the bottom A is a disaster — 125 cents, more than a semitone. Being 2 Hz sharp on the top A is 1 cent, and inaudible. The cent fixes that: one twelve-hundredth of an octave, so a semitone is 100 cents everywhere on the instrument.

For scale, roughly: 1 cent is imperceptible; 5 to 6 cents is about the limit of what a trained listener resolves on sustained tones played one after the other; 10 cents is clearly audible as beating when two notes sound together; 20 cents sounds wrong to almost anyone; and 100 cents is a semitone, a different note.

Beating is the practical version of this. Two notes a few cents apart produce a pulse at the difference of their frequencies — two tones at 440 and 441 Hz beat once a second, which is exactly how a piano tuner sets a tempered fifth: not by hearing the pitch, but by counting the beats per second and matching a target.

The notes an instrument plays without being asked

A vibrating string or air column does not produce one frequency. It produces the fundamental plus a series of whole-number multiples, and the mix of those partials is most of what makes a violin sound unlike a flute at the same pitch.

The harmonic series is pure by construction — the third harmonic is exactly three times the fundamental, which is exactly a pure fifth above the second. So an instrument sounding one note is already producing pure intervals, while the keyboard beside it plays tempered ones. That collision is the real reason equal temperament has a cost.

Two things stand out. The fifth harmonic is 13.69 cents flat of the nearest key — the same figure as the major third error, because it is the major third error, arriving from the other direction.

And the seventh harmonic is 31 cents flat of anything a keyboard can play — nearly a third of a semitone. It is a real, consonant interval that the twelve-note system has no name and no key for. Blues and jazz intonation reaches for it constantly, brass players can hit it as a natural partial, and a piano cannot approach it. When a sung or bent note sounds right in a way the written pitch does not, this is frequently what is happening.

Related calculators

Other logarithmic and ratio-based scales:

More in Conversion, or browse all calculators.

Sources and methodology

The tuning equation and the note numbering are specified rather than conventional, so those come from the MIDI specification and ISO 16. The just-intonation ratios and the cent itself are older, and the Ellis translation of Helmholtz is where the cent enters English-language acoustics — which is also where the divergences this page tabulates were first set out numerically.

• ISO 16:1975 — Acoustics: standard tuning frequency (standard musical pitch)International Organization for Standardization · verified 2026-09-07 · That A4 = 440 Hz is a standardised convention adopted in the twentieth century rather than a physical or acoustic constant — the single fact the reference-pitch control on this page exists to expose
• MIDI 1.0 Detailed Specification — note numbers and the tuning equationMIDI Manufacturers Association · verified 2026-09-07 · The mapping of note number 69 to A4, and the equal-tempered relation f = 440 × 2^((n−69)/12) that every frequency here is computed from
• Acoustical Society of America — scientific pitch notation and the definition of the centAcoustical Society of America · verified 2026-09-07 · The octave-numbering convention in which middle C is C4, and the cent as one twelve-hundredth of an octave on a logarithmic scale
• Helmholtz, On the Sensations of Tone (Ellis translation, with the cent system)Alexander J. Ellis / Internet Archive · verified 2026-09-07 · The origin of the cent as a unit, the just-intonation integer ratios used throughout, and the documented divergence between pure intervals and tempered ones