Labeling pitches

Pitches are often labeled using scientific pitch notation or some combination of a letter and a number representing a fundamental frequency. For example, one might refer to the A above middle C as "A4" or "A440." However, there are two problems with this practice. First, in standard Western equal-temperament, the notion of pitch is insensitive to spelling: the description "G4 double sharp" refers to the same pitch as "A4." Second, human pitch perception is logarithmic with respect to fundamental frequency: the perceived distance between the pitches "A220" and "A440" is the same as the perceived distance between the pitches "A440" and "A880."

To avoid these problems, music theorists sometimes represent pitches using a numerical scale based on the logarithm of fundamental frequency. For example, one can adopt the widely used MIDI standard to map fundamental frequency f to a real number p as follows

p = 69 + 12\times\log_2 { \left(\frac {f}{440} \right) }

This creates a linear pitch space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1, and A440 is assigned the number 69. Distance in this space corresponds to musical distance as measured in psychological experiments and understood by musicians. The system is flexible enough to include "microtones" not found on standard piano keyboards. For example, the pitch halfway between C (60) and C♯ (61) can be labeled 60.5.