Frequencies, like amplitudes, are often described on a logarithmic scale, in order to emphasize proportions between frequencies, which usually provide a better description of the relationship between frequencies than do differences between frequencies. The frequency ratio between two musical tones determines the musical interval between them.
The Western musical scale divides the
octave (the musical interval associated with a ratio of 2:1) into
twelve equal sub-intervals, each of which therefore corresponds to a ratio
of
. For historical reasons this sub-interval is called a
half step.
A convenient logarithmic scale for pitch is simply to
count the number of half-steps from a reference pitch--allowing fractions to
permit us to specify pitches which don't fall on a note of the Western scale.
The most commonly used logarithmic pitch scale is
MIDI, in which the pitch
69 is assigned to the frequency 440, the A above middle C. To convert between
MIDI
pitch and frequency in cycles per second, apply the formulas:
Although MIDI itself (a hardware protocol which has unfortunately insinuated itself into a great deal of software design) allows only integer pitches between 0 and 127, the underlying scale is well defined for any number, even negative ones; for example a "pitch" of -4 is a good rate of vibrato. The pitch scale cannot, however, describe frequencies less than or equal to zero. (For a clear description of MIDI, its capabilities and limitations, see [Bal03, ch.6-8]).
A half step comes to a ratio of about 1.059 to 1, or about a six percent increase in frequency. Half steps are further divided into cents, each cent being one hundredth of a half step. As a rule of thumb, it takes about three cents to make a clearly audible change in pitch--at middle C this comes to a difference of about 1/2 cycle per second.