TWENTIETH- - Synapse Music

152 Form in Twentieth-Century Music
PROPORTION:THE GOLDEN MEAN
Some twentieth-century composers and writers have been interested in the "golden mean" or
"golden section," a proportion used for centuries in art and architecture to obtain aesthetically
pleasing designs. To understand this ratio, consider a line ac with line segments ab and be:
a b c
If the proportion of ab to be is the same as the proportion of be to the whole line, then ac is
segmented according to the golden mean. This relationship can be expressed as:
ab bc
bc ac
The resulting fraction is about .618.
Integers (whole numbers) that approximate the golden mean can be generated by
means of a Fibonacci sequence, an endless series of numbers in which each number is the
sum of the previous two. The farther you go in the sequence, the closer you get to the true
value of the golden mean:
Inlegers
Ratios:
2 3
.5. .67 .6
5
.625
8
.615
13
.619
2 1 34 etc.
The most obvious way that this ratio can be used musically is in the proportions of a musical
form. For example, the beginning of "Minor Seconds, Major Sevenths," from Bart6k's
Mikrokosmos (BUR, p. 483), could be subdivided in this way:
m. 8 = Strong cadence; first whole-note chord.
m. 21 = Strong cadence; first appearance of "glissando."
m.2 l = End of long accelerando and of first main section.
There is some evidence that Bartok used the golden mean not onl y in formal proportions
but in other aspects of his music as well ,7 and this is also true, if to a lesser extent, of some
other twentieth-century composers.8
NONTHEMATIC DELINEATORS OF MUSICAL FORM
The primary determinants of musical foml in tonal music were tonality and theme, with
contrast of tonalities being a generally stronger force than contrast of themes. The decline
in tonality as an organizing force has often led to a greater reliance on thematic contrast;
but in many pieces, themes, in the sense of melodies, playa small or nonex istent part. The
M
most obvious example is electronic music, where texture, register, dynamics, and especial
Iy timbre are usually more important as shaping elements than themes are. Rhythmic activity
is another organizing factor, as in Bart6k's "Increasing-Diminishing" from the
Mikrokosmos ( 1937), where rhythmic activity creates an arch form. Another work in which
nonthematic elements are important in delineating the form is the second movement of
Webern's Concerto, Op. 24 (1934) (K-G, p. 502).
The ABCB form of the first movement of Alvo Part's Collage on B-A -C-H (1964) is delineated
by nonthematie means. The A section (see Example 7-3a) begins with a repealed Bb4
.618