![]() ![]() Eric Weisstein's Treasure Trove of Music. "Equal Temperament - from Eric Weisstein's Treasure Trove of Music". (A 108-key piano that extends from C 0 to B 8 was first built in 2018 by Stuart & Sons. The normally included 88 keys have been numbered 1–88, with the extra low keys numbered 89–97 and the extra high keys numbered 98–108. Keys shaded gray are rare and only appear on extended pianos. Values in bold are exact on an ideal piano. The following equation gives the frequency f of the n th key, as shown in the table: f ( n ) = ( 2 12 ) n − 49 × 440 Hz This deviation from equal temperament is called the Railsback curve. To compensate for this, octaves are tuned slightly wide, stretched according to the inharmonic characteristics of each instrument. On an actual piano the ratio between semitones is slightly larger, especially at the high and low ends, where string stiffness causes inharmonicity, i.e., the tendency for the harmonic makeup of each note to run sharp. This list of frequencies is for a theoretically ideal piano. ![]() For other tuning schemes refer to musical tuning. To go from A 4 to C 5 (which is a minor third), multiply 440 three times by the twelfth root of two (or once by the fourth root of two, approximately 1.189207). To go from A 4 to B 4 (up one whole tone, or two semitones), multiply 440 twice by the twelfth root of two (or once by the sixth root of two, approximately 1.122462). For example, to get the frequency one semitone up from A 4 (A ♯ 4), multiply 440 by the twelfth root of two. Since every octave is made of twelve steps and since a jump of one octave doubles the frequency (for example, the fifth A is 440 Hz and the sixth A is 880 Hz), each successive pitch is derived by multiplying (ascending) or dividing (descending) the frequency of the previous pitch by the twelfth root of two (approximately 1.059463). This is a list of the fundamental frequencies in hertz (cycles per second) of the keys of a modern 88-key standard or 108-key extended piano in twelve-tone equal temperament, with the 49th key, the fifth A (called A 4), tuned to 440 Hz (referred to as A440).
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