Mathematical Structures in John Cage’s
Sonatas and Interludes for Prepared Piano
Throughout the twentieth century, many composers sought to find ways to
organize the new techniques and sounds they incorporated into their music, particularly
in their atonal compositions. As a student of Arnold Schoenberg during Schoenberg’s
tenure at UCLA that began in 1936, American composer John Cage had taken
Schoenberg’s classes dealing with harmony and structure in music. However, as Cage
was primarily a composer of percussion music at that time, he began to see a need for a
musical structure that could be used when harmony was not immediately relevant.
Therefore, in contrast to composing using a structure based on harmony (or pitch, in the
case of the twelve-tone method), Cage developed a method of composition using
structures based on the durations of sounds and silences. The decision to use these kinds
of structures stemmed from Cage’s studies of eastern music and their rhythmic structures
called tala1, and his own ideas about sound that he outlined in his 1949 article
“Forerunners of Modern Music.” According to Cage, “Sound has four characteristics:
pitch, timbre, loudness, and duration. The opposite and necessary coexistent of sound is
silence. Of the four characteristics of sound, only duration involves both sound and
silence. Therefore, a structure based on durations (rhythmic: phrase, time lengths) is
correct (corresponds to the nature of the material), whereas harmonic structure is
1
Cage, John. Silence: Lectures and Writings by John Cage. Cambridge: The M.I.T. Press, 1961, 63.
incorrect (derived from pitch, which has no being in silence).”2 In accordance with this
new premise that structure should be based on duration rather than harmony, Cage was
then able to define structure in music as “its divisibility into successive parts from
phrases to long sections. Form is the content, the continuity. Method is the means of
controlling the continuity from note to note. The material of music is sound and silence.
Integrating these is composing.”3 In his early compositions, these definitions took the
form of structures based on macro-/microcosmic rhythmic proportions and what Cage
called “square-root form.” Both of these rhythmic structures can be seen in the Sonatas
and Interludes for Prepared Piano, his most mature work written using these structures
during his early compositional period that spans from 1933 – 1948.
Cage’s Sonatas and Interludes for Prepared Piano comprise sixteen sonatas and
four interludes. The sonatas are organized into four groups of four, with interludes
placed intermittently, resulting in a palindromic arrangement of groups of sonatas and
interludes (Figure 1).
Sonatas
I – IV
First
Interlude
Sonatas
V–
VIII
Second
Interlude
Third
Interlude
Sonatas
XI –
XII
Fourth
Interlude
Sonatas
XIII –
XVI

Figure 1. Palindromic arrangement of the Sonatas and Interludes.
Most of the sonatas are written in a binary form similar to the form of Domenico
Scarlatti’s keyboard sonatas, with the exception of Sonatas IX, X, and XI which are in
ABBCC, AABBC, and AABCC ternary forms, respectively. The interludes lack any
2
Ibid, 63.
3
Ibid, 62.
3
universally unifying form—the First and Second Interludes are through-composed and
lack any structural repetitions, while the Third and Fourth Interludes are in four-part
AABBCCDD forms.4
While Cage uses binary and ternary forms—forms that are distinctly indicative of
Western music because of their harmonic expectations—he uses them not with respect to
the harmonic structure they normally reflect, but with respect to durations manifested as
rhythmic units. According to Cage in his 1958 speech “Composition as Process: I.
Changes,” the rhythmic structure of the Sonatas and Interludes for Prepared Piano are
“as hospitable to non-musical sounds, noises, as it [is] to those of the conventional scales
and instruments. For nothing about the structure was determined by the materials which
were to occur in it; it was conceived, in fact so that it could be as well expressed by the
absence of these materials as by their presence.”5 While Cage had been exploring this
new structure based on pre-compositionally conceived rhythmic proportions and another
rhythmic structure that he called “square-root form” in his earlier works (namely
Metamorphosis, Five Songs, and Imaginary Landscape No. 16), the Sonatas and
Interludes remain his most mature and rhythmically complex example of these methods,
because unlike his previous compositions, the proportions involved in the composition of
the Sonatas and Interludes involve fractions that result in meter changes, asymmetrical
phrases, and obscured rhythmic patterns.
4
Pritchett, James. The Music of John Cage. Cambridge: Cambridge University Press, 1993, 30.
5
Cage, Silence, 19-20.
6
Nicholls, David, ed. The Cambridge Companion to John Cage. Cambridge: Cambridge University Press,
2002, 67.
4
A less obscured example of fractional rhythmic proportions occurs in Sonata V
(Figure 2), the first sonata of the second group of four sonatas. Sonata V is divided into
nine-measure structural units designated by the double barlines that occur after every nine
measures. The A section of the sonata (up to the first repeat) contains two nine-measure
units. The B section of the sonata contains two and a half nine-measure units. The onehalf unit is the result of the last four measures, three of which are in 2/2 meter and the last
measure that is in 3/2 meter (which is equivalent to one and a half measures of 2/2 meter)
resulting in four and a half measures, or half of a nine-measure unit. Observing repeats,
this results in a [ 2 : 2 : 2 ½ : 2 ½ ] macrocosmic, or large form, proportion.
The [ 2 : 2 : 2 ½ : 2 ½ ] proportion is also reflected microcosmically within each
nine-measure unit. Beginning with the first nine-measure unit, the first two measures
comprise the first rhythmic motive that is repeated as two units of two measures each.
These units are followed by an extension of the original two-measure unit that is now two
and a half measures long, followed by a unit of one measure and another of one and a
half measures. This results in a microcosmic proportion of [ 2 : 2 : 2 ½ : 2 ½ ] – the same
proportion that applies macrocosmically to the entire sonata. The rest of the ninemeasure units in the sonata may be proportioned similarly.
Not all of the sonatas, however, can be analyzed as readily. Sonata I (Figure 3)
poses a difficulty to the structural analysis through its use of changing meter that occurs
throughout. Because the structure is based on duration, that is, lengths of time, each
meter that is different from the beginning meter must be converted to the beginning meter
before analysis can occur so that the durations are divided equally. (Figure 4). One
structural unit in Sonata I is equivalent to seven measures of 2/2 meter. The first seven
5
Figure 2. Structural proportions in Sonata V.7
7
Cage, John. Sonatas and Interludes for Prepared Piano. New York, Henmar Press, 1960, 10.
6
measure unit occurs during the first seven measures of 2/2 meter. However, the second
unit occurs in the next five measures as a result of the changing meter. The one measure
of 7/4 meter converts to one and three-fourths measures of 2/2 meter. The following
measures of 6/4 meter converts to one and a half measures of 2/2 meter. The measures of
9/8 meter, if first converted to eighth notes are equivalent to one and one eighth measures
of 2/2 meter. Therefore, adding all the converted measure lengths results in a sevenmeasure unit of 2/2 meter (1 ¾ measures + 1 ½ measures + 1 ½ measures + 1 1/8
measures + 1 1/8 measures = 7 measures of 2/2 meter = 1 rhythmic unit). The B section
of the sonata is written in 2/2 meter until the last seven-measures, which are in 2/4 meter.
When these measures are converted to 2/2 meter, this results in three and one-half
measures of 2/2 meter, or half of a seven-measure unit. Therefore, the macrocosmic
proportion of Sonata I is [ 2 : 2 : 1 ½ : 1 ½ ].
The microcosmic structure of Sonata I, while nearly completely obscured by the
changing meter in the A section, is still present. In the first two measures of 2/2 meter,
the first two measures are a unit of two measures, broken into rhythmic motives of one
and one-half measures and three-fourths measure. The following two measures mimic
the rhythmic structure of the first two measures. The last three measures of the first
rhythmic unit are broken into two one and one-half measure units (Figure 3). Additively,
this results in a microcosmic rhythmic proportion of [ 2 : 2 : 1 ½ : 1 ½ ].
The following five measures obscure the microcosmic structure so much that
music theorist David W. Bernstein writes in his article “Music I: to the late 1940s”
printed in The Cambridge Companion to John Cage, “the following 7/4 and 6/4 measures
do not parse into units of 1¼ and ¾ (durations of five and three quarter notes in length).
7
Figure 3. Structural proportions in Sonata I.8
8
Cage, Sonatas and Interludes for Prepared Piano, 3.
8
Meter
7/4
Conversion units
for one measure
4 quarter notes, or 8
eighth notes
7 quarter notes
6/4
6 quarter notes
9/8
9 eighth notes
9 (eighth notes in
9/8) ÷ 8 (eighth
notes in 2/2) = 1.125
1 ¾ measures of 2/2
meter per measure
of 7/4
1 ½ measures of 2/2
meter per measure
of 7/4
1 1/8 measures of
2/2 meter per
measure of 9/8
2/4
2 quarter notes
2 (quarter notes in
2/4) ÷ 4 (quarter
notes in 2/2) = 0.5
½ measure of 2/2
meter per measure
of 2/4
2/2
Analysis
----------7 (quarter notes in
7/4) ÷ 4 (quarter
notes in 2/2)= 1.75
6 ÷ 4 = 1.5
Conversion to 2/2
meter
-----------
Figure 4. Rhythmic analysis of the A section of Sonata I.
The microstructure begins to re-emerge in the concluding measures of the “a” section
(mm. 10 – 12) and appears unambiguously at the beginning of the “b” section (mm. 25ff).
Cage was quite willing to deviate from his preconceived rhythmic plan, or in his own
words ‘play with and against the clarity of the rhythmic structure’ for expressive
purposes.”9 It seems unquestionable that Cage has used the changing meter in this
Sonata for “expressive purposes”; however, it is unlikely that he completely deviated
from his preconceived rhythmic plan. If the 1¼ measures and ¾ measures are used
additively in the first seven measures to result in the same proportion as the macrocosmic
structure, why could the measures containing metrical changes not be approached in the
same way? Measure 8 to the downbeat of m. 9 comprise what converts to two measures
of 2/2 meter; the rest of m. 9 through the third beat of m. 10 comprises another two
9
Nicholls, 83.
9
measures of 2/2 meter. From beat three of m. 10 to the end of m. 12 comprise three
measures of 2/2 meter that may be divided between the tied E and F in m. 11 to yield two
one and a half measure units of 2/2 meter, thus additively giving the proportion [ 2 : 2 : 1
½ : 1 ½ ] the same as the macrocosmic proportional structure.
The B section of Sonata I begins by using the same proportions as the first seven
measure unit of the A section; however, the last seven measures of the sonata that appear
in 2/4 meter are, in fact, only a half unit after conversion to 2/2 meter, thus yielding the
[2 : 2 : 1 ½ : 1 ½ ] macrocosmic proportion. It may be interesting to note, however, that
into this half unit, Cage manages to place another statement of the microcosmic
proportion equivalent to the first seven-measure unit, but in half the time.
In addition to skillfully applying a macro-/microcosmic structure to the Sonatas
and Interludes for Prepared Piano, John Cage also determined the structure of the
Sonatas and Interludes using what he called “square-root form”, a form that he began
using in his early percussion works, especially the in the Constructions. In this form, the
number of measures in one rhythmic unit equals the square root of the number of
measures in the entire piece (Equation 1). As Cage subtly applied the macro/microcosmic structure to the sonatas previously examined, he simultaneously applied
square-root form.
[One rhythmic unit, in measures] = √[Total measures in the piece, including repeats]
Equation 1. John Cage’s “Square-root form.”
Sonata V, which uses a [ 2 : 2 : 2 ½ : 2 ½ ] macro-/microcosmic structure is
divided into nine measure units. Counting all of the measures, and counting the last
10
measure as 1 ½ measures since it occurs in 3/2 rather than in 2/2, the total number of
measures in the piece is 40 ½. But since each section is repeated, the total number of
measures in the sonata must be multiplied by two, yielding 81 total measures in Sonata
V. Thus, taking the square root of the total number of measures in the sonata with
repeats gives 9 measures, the length in measures of one rhythmic unit.
Similarly, Sonata I, which uses a [ 2 : 2 : 1 ½ : 1 ½ ] macro-/microcosmic
structure divided into 7 measure units, also follows square root form. Counting measures
made by converting to 2/2 meter as explored previously, the total number of measures in
the sonata is forty-nine, which yields the square-root unit of seven measures – the same
length as the original rhythmic unit.
This rhythmic structure extends to the Interludes, as does the structure based on
macro-/microcosmic proportions. For example, the Fourth Interlude is based on a
rhythmic unit of 8 ½ measures of 4/4 meter which occur in the proportion [1 : 1 : 1 : 1 : 1
: 1 : 1 ¼ : 1 ¼]. Observing repeats and converting other time signatures to 4/4 meter,
this yields 72 ¼ measures of 4/4 meter. Taking the square root of 72 ¼ yields 8 ½ or one
rhythmic unit. While some sonatas and interludes seem rhythmically obscure due to
changing meter and expressive style, all of the sonatas and interludes in the work are
based on both the structures determined by macro-/microcosmic structure and by square
root form.
While the Sonatas and Interludes for Prepared Piano are quite rhythmically
complex, the rhythmic proportions are not the only factor that contributes to their
continuity and musical expression. In many of his earlier compositions, Cage sought to
create a new kind of structure that was based solely on durations rather than on harmony;
11
nevertheless, the Sonatas and Interludes are not without some subtle tonal implications.
For example, Sonata IV (Figure 5) seems to center on middle-A—one of the few notes,
especially in this register, that Cage left unprepared. Similarly, Sonata VI seems to center
on B until the final measures, which seems to resolve to e minor (Figure 6). Sonata X
centers on B, ending more ambiguously with a sustained open fifth from B to F-sharp
(Figure 7), while Sonata XII, which also centers around B, ends more discernibly in d
minor (Figure 8). The final sonata, Sonata XVI, is (not surprisingly as the final sonata
movements are supposed to represent tranquility) the most clearly tonal, remaining
loosely in G major for the entire movement, as evidenced by the G major chords, G
pedals, and F-sharp accidentals that occur throughout the movement (Figure 9).
Figure 5: Sonata IV centers on the unprepared middle-A10
10 Cage, Sonatas and Interludes for Prepared Piano, 6.
12
Figure 6: E minor implications in the final measures of Sonata VI11
Figure 7: B minor implications in the final measures of Sonata X12
Figure 8: D minor implications in the final measures of Sonata XII13
11
Ibid, 11.
12
Ibid, 22.
13
Ibid, 25.
13
Figure 9: G major implications in Sonata XVI (m. 24ff)14
The tonal implications of these sonatas are vitally important, as they show that
John Cage’s new system of organizing sound with respect to rhythm rather than with
respect to harmony is equally accommodating of tonal music as it is of non-tonal music.
According to John Cage, the other new methods of composition beginning to be explored
during the post-war era (specifically, serialism and neoclassicism) cannot claim to be as
accommodating of both constructs. In “Forerunners of Modern Music” Cage comments
that
Atonality is simply the maintenance of an ambiguous tonal state of affairs.
It is the denial of harmony as a structural means. The problem of a
composer in a musical world in this state is to supply another structural
means, just as in a bombed-out city the opportunity to build again exists.
This way one finds courage and a sense of necessity.
14
Ibid, 32.
14
Neither Schoenberg nor Stravinsky did this. The twelve-tone row does not
offer a structural means; it is a method, a control, not the parts, large and
small, of a composition, but only of the minute, note-to-note procedure. It
usurps the place of counterpoint, which, as Carl Ruggles, Lou Harrison,
and Merton Brown have shown, is perfectly capable of functioning in a
chromatic situation. Neo-classicism, in reverting to the past, avoids, by
refusing to recognize, the contemporary need for another structure, gives a
new look to structural harmony. This automatically deprives it of the
sense of adventure, essential to creative action.15
The twelve-tone method and the methods associated with neoclassicism are only
methods, or a means of controlling the content from note to note rather than a structure.
The reason that a structure based on durations is so accommodating of both tonal and
non-tonal constructs is because it is the only structure that is able to organize all sound
and its necessary counterpart, silence, regardless of its origin or the method by which its
note-to-note continuity is composed. John Cage’s Sonatas and Interludes for Prepared
Piano is truly the masterwork that employs a durational structure as it not only
incorporates percussive sounds and silences that otherwise would not be able to be
incorporated into the structure of the work, but uses tonal means simultaneously, thus
proving the validity of the new structure for all music.
15
Cage, Silence, 63-64.