Pitch space

In music pitch space is pitch relations, ie nearness or farness, represented through geometric models, most often multidimensional, how near or far pitches are from each other. Cognitive psychologists including Longuet-Higgins (1978) and Shepard (1982), and composers and theorists including Weber (1824), Riemann, and Schoenberg (1954) created models of pitch space. There are generally at least two dimensions, one for pitch class and one for register (ie, the specific pitch), but there may be any number. (Lerdahl, 1992)

Modulatory space is the pitch space within which modulation is possible. For twelve tone equal temperament, this includes only the twelve pitch classes.

The circle of fifths is one representation of pitch space, first proposed geometrically (see: Pythagoras) by Johann David Heinichen (1728), though he included the relative minor (thus the circle clockwise would read C, a, G, e...) (Lerdahl, 2001). The current major on the outside relative minor on the inside format was proposed by David Kellner (1737). M.W. Drobisch (1855) was the first to suggest a helix (ie the spiral of fifths) to represent octave equivalency and reoccurance (Lerdahl, 2001). Shepard (1982) uses a double helix of two wholetone scales over a circle of fifths which he calls the "melodic map" (Lerdahl, 2001). Michael Tenzer suggests its use for Balinese gamelan music since the octaves are not 2:1 and thus there is even less octave equivalency than in western tonal music (Tenzer, 2000). See also chromatic circle.

Weber's "regional chart" centered on C major is:

d#	F#	f#	A	a	C	c
g#	B	b	D	d	F	f
c#	E	e	G	g	Bb	bb
f#	A	a	C	c	Eb	eb
b	D	d	F	f	Ab	ab
e	G	g	Bb	bb	Db	db
a	C	c	Eb	eb	Gb	gb

Lower case letters indicate minor key, uppercase major. This was first proposed by Vial (1767) (later Weber, Riemann, Schoenberg), the advantage over the circle of fifths being that it represents both relative and parallel major. (Lerdahl, 2001)

The use of a lattice was first proposed by Euler (1739) to model just intonation using an axis of perfect fifths and another of major thirds (Lerdahl, 2001). James Tenney argues for multidimensional lattices, especially for just intonation systems, which contain a dimension for every pitch axis used (Tenney, 1983). Thus if a justly tuned system is based on the octave and fifths it would contain only two dimensions. W. A. Mathieu uses this perfect fifths and major thirds also (Mathieu, 1997) (see sargam).

Riemann's Tonnetz:

A#	—	E#	—	B#	—	FX	—	CX	—	GX
|		|		|		|		|		|
F#	—	C#	—	G#	—	D#	—	A#	—	E#
|		|		|		|		|		|
D	—	A	—	E	—	B	—	F#	—	C#
|		|		|		|		|		|
Bb	—	F	—	C	—	G	—	D	—	A
|		|		|		|		|		|
Gb	—	Db	—	Ab	—	Eb	—	Bb	—	F
|		|		|		|		|		|
Ebb	—	Bbb	—	Fb	—	Cb	—	Gb	—	Db

Perfect fifths are the horizontal axis, major thirds the vertical. First proposed by Euler, later used, not always in just intonation, by Hermann von Helmholtz (1863/1885), Arthur von Oettingen (1866), Renate Imag (1970), Longuet-Higgins (1962), Shepard (1982) "harmonic map"

Harry Partch's "Tonality Diamond" is similar:

<math> \begin{matrix} \; & \frac{3}{3} & \; \\ \frac{4}{3} & \; & \frac{3}{2} \\ \; & \frac{2}{2} & \; \end{matrix}<math>

3-limit just intonation

Deutsch and Feroe (1981), and Lerdahl and Jackendoff (1983) use a "reductional format" representing pitch relations by "alphabets" or hierarchy of levels such as the chromatic, diatonic, and triadic. Lerdahl's levels include the octave, perfect fifth, major triad, diatonic scale, and the chromatic scale:

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

Db

D

Eb

E

F

F#

G

Ab

A

Bb

B

C

(Lerdahl, 1992)

According to David Kopp (2002), "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." (p.1)

The matrices used in the twelve tone technique are not representations of pitch space as nearness nor farness is not indicated, or even possible since one may not move freely about.

See also

• Vowel space
• Emancipation of the dissonance
• Diatonic set theory
• Color space

External link

• The James Tenney Project: excerpts from John Cage and the Theory of Harmony (http://www.music.mcgill.ca/~gems/tenney/theory.html)
• TonalSoft.com: Lattices (http://tonalsoft.com/monzo/lattices/lattices.htm) Text and diagrams © 1998 by Joseph L. Monzo

Sources

• Lerdahl, Fred (1992). Cognitive Constraints on Compositional Systems, Contemporary Music Review 6 (2), pp. 97-121.
• Lerdahl, Fred (2001). Tonal Pitch Space, pp. 42-43. Oxford: Oxford University Press. ISBN 0195058348.
• Tenney, James (1983). John Cage and the Theory of Harmony.
• Tenzer, Michael (2000). Gamelan Gong Kebyar: The Art of Twentieth-Century Balinese Music. Chicago: University of Chicago Press. ISBN 0226792811.
• Mathieu, W. A. (1997). Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression. Inner Traditions Intl Ltd. ISBN 0892815604.
• Kopp David, (2002). Chromatic Transformations in Nineteenth-Century Music. Cambridge University Press. ISBN 0521804639.