Pitch space

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.

This has led theorists such as Moritz Wilhelm Drobisch (1846) and Roger Shepard (1982) to model pitch relations using a helix.

Other theorists, such as Leonhard Euler (1739), Hermann von Helmholtz (1863/1885), Arthur von Oettingen (1866), Hugo Riemann (not to be confused with mathematician Bernhard Riemann), and Christopher Longuet-Higgins (1978) have modeled pitch relationships using two-dimensional (or higher-dimensional) lattices, under the name of Tonnetz.

The idea of pitch space goes back at least as far as the ancient Greek music theorists known as the Harmonists[citation needed].

And we use a diagram so that, for students of the subject, matters which are hard to grasp with the hearing may appear before their eyes" (Bacchius, in Franklin, Diatonic Music in Ancient Greece).

The use of a lattice was proposed by Euler (1739) to model just intonation using an axis of perfect fifths and another of major thirds.

Similar models were the subject of intense investigation in the 19th century, chiefly by theorists such as Oettingen and Riemann (Cohn 1997).

Moritz Wilhelm Drobisch (1846) was the first to suggest a helix (i.e. the spiral of fifths) to represent octave equivalence and recurrence (Lerdahl, 2001), and hence to give a model of pitch space.