The word root, and the harmonic metabolism it helps to describe, plays a central role in how musicians think about musical structure. Vaguely implicit in some early writers' descriptions of intervals and scales, it awaited Rameau's discussions (1722/1971) of the fundamental bass to become explicit. Since then, music theorists have sought to explain its perceptual nature and causes. Their theories usually turn on some version of physical weightings, a root being the pitch class more powerfully reinforced than its companion pitch classes. After representative versions of influential explanations are reviewed, and their generic shortcomings are noted, Ernst Terhardt's "virtual pitch" theory is recognized as uniquely reasonable; it embodies a condition of pattern perception rather than physical reinforcement, thereby skirting a principal flaw of past theories. And yet, a troublesome paradox surfaces, regardless of which conceptualization one favors: empirical studies of interval perception have fallen short of confirming the phenomenal reality our concepts describe so confidently. In an attempt to outflank these empirical/phenomenal clashes, a scheme of pitch/time interaction, or "vectoral dynamics", is outlined. Its consistency with the linear perspective of vision is noted, and the model is applied to the opening of a Wolf song and to a painting by Titian.

[Footnotes]

[Footnotes]
1
Articles IV and V, Book I, Chapter 3 of Rameau (1922/1971).
2
"Root," in The New Grove Dictionary of Music and Musicians (vol. 16, p. 183).
3
Rameau (1922/1971, Book IV, Chapter 10
a 3rd added below the same note (Génération Harmonique, Chapt. 9).
Hindemith resurrects the double emploi in his Tradi- tional Harmony (pp. 48-49).
4
Riemann gave Heinrich Koch (1811)
Rameau (1922/1971, Book II, especially in Chapter 10)
by Riemann of parallel clangs can be found in his Vereinfachte Harmonielehre, 1893 (Harmony Simplified, 1896). Also see Mickelsen (1977, pp. 63-65).
5
Hindemith (1942, p. 92)
6
Problems, Book I, of Pseudo-Aristotle, and Makund Lath's treatise on the music of ancient India
(wu sheng) of Kung, Shang, Chiao, Chih, and Yut and two elaborative tones (Pien tones).
7
Rameau (1922/1971, Book I, Chapter 7)
8
F. Kreuger's (1910)
10
Parncutt (1988)
(Parncutt, 1988, p. 69).
13
Terhardt's own study (1977, pp. 386-387 and 1984, p. 291)
14
Van Dyke Bingham's (1910) pioneer study.
16
Schoenberg (1954, p. 1):
Krenek (1939, p. 108):
17
Allen Forte's (1973, p. 15)
18
(1974, pp. 1063, 1066; 1984, p. 288)
19
Ritsma (1962, 1963, 1967) concludes that nos. 3, 4, and 5 are most effective in conveying pitch location
Smoorenburg (1970),
20
Neisser (1967. pp. 186-193) uses the term schema for what I mean here by template.
21
Allport (1955, pp. 497, 635) uses the term regenerative loop in discussing this kind of circuitry.
22
Thomson (1991, chap. 10).
23
Hall and Peters (1981, p. 509)
24
Terhardt's model (1989, pp. 82-86).

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