Frequency to Note Converter

Pitch relates to frequency logarithmically, not linearly, which is why this conversion takes a base-2 logarithm rather than a difference. In twelve-tone equal temperament every semitone is the same frequency ratio, the twelfth root of 2, about 1.059463; twelve such steps multiply to exactly 2, so doubling a frequency always raises pitch by one octave. The expression n = 12 x log2(f / refA) returns the exact, usually fractional, number of semitones from the reference. Rounding it identifies the nearest chromatic note; the leftover fraction times 100 gives the deviation in cents, where one cent is a hundredth of a semitone and a twelve-hundredth of an octave. MIDI note numbers sit on the same grid, with A4 fixed at 69 and middle C at 60. Players generally accept plus or minus 5 cents as in tune, while piano and orchestral tuners work nearer 2 cents. The reference is not universal: 440 Hz is the ISO figure, many European orchestras tune to 442 or 443 Hz, and historically informed baroque ensembles commonly use 415 Hz, roughly a semitone lower. Two limits are worth noting. Equal temperament is only one system, so a cents readout is not a verdict on musical correctness; a just-intonation major third sits about 14 cents below its equal-tempered equivalent. And real instruments are not single sine waves. Piano strings are inharmonic, and stretched tuning deliberately sharpens the top octave by 20 to 30 cents. The wavelength figure assumes 343 m/s, which holds for dry air near 20 degrees C.

Standard pitch is a relatively recent agreement. Surviving organ and tuning-fork evidence shows European A varying from roughly 380 to 480 Hz between the sixteenth and eighteenth centuries. France legislated the first widely adopted standard in 1859, setting a diapason normal of A = 435 Hz on the advice of a commission that included Berlioz and Rossini. Rising orchestral pitch prompted further work, and an international conference held in London in 1939 recommended A = 440 Hz. The International Organization for Standardization later codified that figure as ISO 16, which remains the reference cited on tuners and in instrument specifications today. The mathematics behind the conversion is older than the standard. The twelfth root of 2 was computed independently by Zhu Zaiyu in Ming China, whose work on equal division appeared in 1584, and by Simon Stevin in the Low Countries at about the same time, though equal temperament only displaced meantone and well temperaments across Europe during the nineteenth century. The cent was introduced by Alexander J. Ellis, who set out the 1200-per-octave scale in material he added to his English edition of Helmholtz's On the Sensations of Tone in the 1880s, giving comparative musicologists a unit fine enough to describe non-Western scales. Integer note numbering arrived with the MIDI specification agreed by synthesizer manufacturers in 1983.