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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). [1] [2] Every octave is made of twelve steps called semitones.
The quantity log 2 ( f / 440 Hz ) is the number of octaves above the 440 Hz concert A, or A 4, or a ′ . Multiplying it by 12 gives the number of semitones above 440 Hz (the value is negative if the frequency f is lower in pitch than 440 Hz).
The purpose of this adjustment is to move the 12 notes within a smaller range of frequency, namely within the interval between the base note D and the D above it (a note with twice its frequency). This interval is typically called the basic octave (on a piano keyboard, an octave has only 12 keys).
A Pythagorean tuning is technically both a type of just intonation and a zero-comma meantone tuning, in which the frequency ratios of the notes are all derived from the number ratio 3:2. Using this approach for example, the 12 notes of the Western chromatic scale would be tuned to the following ratios: 1:1, 256:243, 9:8, 32:27, 81:64, 4:3, 729: ...
Notes played in tune with the 12 equal temperament system will be an integer number of half-steps above (positive ) or below (negative ) that reference note, and thus have a frequency of: f = 2 h 12 × 440 Hz {\displaystyle f=2^{\frac {h}{12}}\times 440{\text{ Hz}}\,}
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Therefore, any note and its octaves will generally be found similarly named in musical systems (e.g. all will be called doh or A or Sa, as the case may be). When expressed as a frequency bandwidth an octave A 2 –A 3 spans from 110 Hz to 220 Hz (span=110 Hz). The next octave will span from 220 Hz to 440 Hz (span=220 Hz).
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