Editing Modulatory space (section)

==Circular pitch class space==

[[Image:Pitch class space.svg|thumb|Circular pitch class space]]
The simplest pitch space model is the real line.  In the [[MIDI Tuning Standard]], for example, fundamental frequencies ''f'' are mapped to numbers ''p'' according to the equation

:<math>
p = 69 + 12\log_2 {(f/440)}
</math>

This creates a linear space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1, and [[A440 (pitch standard)|A440]] is assigned the number 69 (meaning [[middle C]] is assigned the number 60).  To create circular '''pitch class space''' we identify or "glue together" pitches ''p'' and ''p'' + 12.  The result is a continuous, circular [[pitch class space]] that mathematicians call '''Z'''/12'''Z'''.