Note Frequency Calculator

Pitch in twelve-tone equal temperament is geometric, not linear. An octave is a frequency doubling, and the octave is divided into twelve equal steps, so one semitone is a ratio of the twelfth root of 2, about 1.059463. That is why the frequency of any note is the reference frequency multiplied by 2 raised to the power of the semitone distance divided by 12, and why finer work uses cents, a logarithmic unit of one hundredth of a semitone and 1200 to the octave. With A4 fixed at 440 Hz the system yields C4 at 261.626 Hz, the lowest piano note A0 at 27.5 Hz, and the top note C8 at 4186.01 Hz. Two derived quantities follow directly: the period is the reciprocal of frequency, and the wavelength is the speed of sound divided by frequency, using about 343 metres per second in air at 20 degrees C. Harmonics are integer multiples of the fundamental and set an instrument timbre. The limitations are worth stating plainly. Equal temperament is a compromise, so only the octave is acoustically pure: its major third is roughly 14 cents wider than the just 5 to 4 ratio and its fifth about 2 cents narrower than 3 to 2. Real instruments are also inharmonic. Piano strings have bending stiffness, so their upper partials run progressively sharp of exact integer multiples, which is why tuners stretch the octaves and why a piano tuned to strict calculated frequencies sounds flat in the treble and sharp in the bass. Wavelength figures shift with conditions, since the speed of sound in air changes by roughly 0.6 metres per second per degree Celsius, so a value computed at 20 degrees C is wrong outdoors on a cold day. Trained listeners can typically detect pitch differences of about 5 to 6 cents in sustained tones, which sets the practical precision worth chasing.

The idea that consonance corresponds to simple frequency ratios goes back to the Pythagorean monochord experiments, but the mathematics of equal division waited on the ability to compute irrational roots. That step was taken twice within about a year: Zhu Zaiyu, a prince of the Ming dynasty, published a calculation of the twelfth root of 2 for tuning purposes in 1584, and the Flemish mathematician Simon Stevin worked out the same division in the Netherlands shortly afterward. Marin Mersenne set out the relationships between string length, tension and frequency in Harmonie universelle in 1636, moving the subject from ratio arithmetic to measurable vibration. Equal temperament spread slowly and unevenly across Europe, tied to keyboard practice and the desire to modulate freely between keys. The reference frequency was standardised far later than the tuning system. Concert pitch varied widely by city and era, with Baroque practice often near A equals 415 Hz and nineteenth-century orchestras drifting upward for brightness. A congress at Stuttgart recommended 440 Hz in 1834, the French government legislated the diapason normal at A equals 435 Hz in 1859, an international conference in London recommended 440 Hz in 1939, and the International Organization for Standardization codified it as ISO 16, first issued in 1955 and revised in 1975. The cent, the unit this calculator uses for fine offsets, was introduced by Alexander John Ellis in the 1880s in his annotated English translation of Hermann von Helmholtz On the Sensations of Tone, precisely so that tunings from different cultures and eras could be compared on one logarithmic scale.