Piano Key Frequency Calculator

Equal Temperament Modern pianos use 12-tone equal temperament (12-TET): the octave is divided into 12 equal semitones. Each semitone ratio = 2^(1/12) ≈ 1.05946. The frequency doubles with each octave. Standard concert pitch: A4 = 440 Hz (adopted as international standard in 1939, London).

The Formula f(n) = 440 × 2^((n - 49)/12) Where n = key number on the piano (A0 = key 1, C8 = key 88, A4 = key 49). Or equivalently: f = A4_Hz × 2^(semitones_from_A4 / 12)

MIDI Note Numbers MIDI (Musical Instrument Digital Interface) uses numbers 0 to 127 for notes. Middle C (C4) = MIDI 60. A4 (440 Hz) = MIDI 69. Formula: f(MIDI) = 440 × 2^((MIDI - 69)/12) MIDI was developed in 1983 as a universal standard connecting synthesizers and computers. The 88 piano keys sit inside that range as MIDI 21 to 108, so a keyboard controller can address 40 notes no piano has.

Historical Tuning Systems Before equal temperament, many systems existed: Just intonation: pure frequency ratios (3/2 = perfect fifth, 5/4 = major third). Sounds pure but only in one key. Meantone temperament: compromise for keyboard instruments, common in Europe from about 1550 to 1800. Well temperament: Bach’s “Well-Tempered Clavier” (Germany, 1722) may have used a form of this. Today’s equal temperament lets you play in any key equally, at the cost of slight impurity in every interval except the octave.

The Physics of Piano Strings String frequency: f = (1/2L) × √(T/μ) L = string length, T = tension, μ = mass per unit length. Higher notes: shorter, thinner strings. Lower notes: longer, heavier strings (wound with copper). Piano tuning is also affected by inharmonicity. Real strings have stiffness, so their upper partials run sharp of exact multiples of the fundamental, and a piano tuned to perfect mathematical octaves sounds flat in the bass and treble. Tuners answer that by stretching the octaves, which is why the top note of a concert grand can sit 30 cents above the number this calculator returns and still be right.

Concert A Variations A4 = 440 Hz: current international standard (ISO 16). A4 = 432 Hz: promoted in some corners of the internet, with no acoustic or physiological advantage behind the claims. A4 = 415 Hz: Baroque pitch, commonly used for period instrument recordings. It is almost exactly a semitone below 440, which is convenient for anyone reading modern parts. A4 = 466 Hz: High Baroque, used by some German organs and orchestras in the 1700s. A4 = 443 to 444 Hz: some European orchestras today, including Berlin and Vienna, for a brighter sound.