Tonal Pitch Class Space
Deutsch and Feroe (1981), and Lerdahl and Jackendoff (1983) use a "reductional format" to represent the perception of pitch class relations in tonal contexts. These two-dimensional models resemble bar graphs, using height to represent a pitch class's degree of importance or centricity. Lerdahl's version uses five levels: the first (highest) contains only the tonic, the second contains tonic and dominant, the third contains tonic, mediant, and dominant, the fourth contains all the notes of the diatonic scale, and the fifth contains the chromatic scale. In addition to representing centricity or importance, the individual levels are also supposed to represent "alphabets" that describe the melodic possibilities in tonal music (Lerdahl, 2001, 44–46). The model asserts that tonal melodies will be cognized in terms of one of the five levels a-e:
| Level a: | C | C | |||||||||||
| Level b: | C | G | C | ||||||||||
| Level c: | C | E | G | C | |||||||||
| Level d: | C | D | E | F | G | A | B | C | |||||
| Level e: | C | D♭ | D | E♭ | E | F | F♯ | G | A♭ | A | B♭ | B | C |
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- (Lerdahl 1992,)
Note that Lerdahl's model is meant to be cyclical, with its right edge identical to its left. One could therefore display Lerdahl's graph as a series of five concentric circles representing the five melodic "alphabets." In this way one could unite the circular representation depicted at the beginning of this article with Lerdahl's flat two-dimensional representation depicted above.
According to David Kopp (2002, 1), "Harmonic space, or tonal space as defined by Fred Lerdahl, is the abstract nexus of possible normative harmonic connections in a system, as opposed to the actual series of temporal connections in a realized work, linear or otherwise."
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