Pitch space

In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships under consideration, the models may be multidimensional. Models of pitch space are often graphs, groups, lattices, or geometrical figures such as helixes. Pitch spaces distinguish octave-related pitches. When octave-related pitches are not distinguished, we have instead pitch class spaces, which represent relationships between pitch classes. (Some of these models are discussed in the entry on modulatory space, though readers should be advised that the term "modulatory space" is not a standard music-theoretical term.) Chordal spaces model relationships between chords.

The simplest pitch space model is the real line. A fundamental frequency f is mapped to a real number p according to the equation

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 middle C is assigned the number 60, as it is in MIDI. 440 Hz is the standard frequency of 'concert A', which is the note 9 semitones above 'middle C'. Distance in this space corresponds to physical distance on keyboard instruments, orthographical distance in Western musical notation, and psychological distance as measured in psychological experiments and conceived 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.

One problem with linear pitch space is that it does not model the special relationship between octave-related pitches, or pitches sharing the same pitch class. This has led theorists such as M. W. Drobish (1855) and Roger Shepard (1982) to model pitch relations using a helix. In these models, linear pitch space is wrapped around a cylinder so that all octave-related pitches lie along a single line. Care must be taken when interpreting these models however, as it is not clear how to interpret "distance" in the three-dimensional space containing the helix; nor is it clear how to interpret points in the three-dimensional space not contained on the helix itself.

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