I pick mathematician! I know that wasn't on your list, Arnold,
but it so happens that an interesting mathematical concept called square root
form is at work in this unusual composition.
As Arnold mentioned, Cage uses the "gamut" technique, with its
pre-determined assortment of melodic notes and chords
that are sounded one at a time without a sense
of progression, in the context of a four movement work,
where there are 22 episodes that are each 22 measures in length.
As you can already surmise, there is nothing traditional about this quartet.
According to Cage, the four movements, "Quietly Flowing," "Slowly Rocking,"
"Nearly Stationary" and "Quodlibet" exhibit a rhythmic structure in which
the following numbers play a role: 2 1/2, 1 1/2, 2, 3,
6, 5, .5, and 1.5. The sum total of these numbers is 22.
For the first movement's structure, Cage added 2 1/2 plus 1 1/2
for a total of 4. He then multiplied this number by 22
for a product of 88. So, the first movement has four episodes that are each
22 measures in length for a total of 88 measures. He follows this
procedure for the 2nd movement by adding the next two numbers
in this sequence - 2 + 3 for a sum of 5.
Five times 22 yields 110 measures of length for this movement,
that is 5 episodes that are 22 measures each. He continues this process
for movements three and four by adding the next two numbers in the sequence,
and multiplying the sum by 22. The end result of the work is 22
episodes that are each 22 measures in length,
hence the term "square-root form." Now, you might be considering
how these numbers affect the expression of this work
which captures the essence of changing seasons,
summer in Paris, fall in America, winter and finally spring. Well, there
is the distinct feeling that the music is slowing down as the movements
proceed from "Quietly Flowing,"
to "Slowly Rocking" and to "Nearly Stationary."
Even though the tempo remains consistently slow
at half-note equals 54 on the metronome,
Cage moves to longer and longer rhythmic values
from one movement to the next and the dynamics remain relatively
quiet. In addition, the square root form
which displays an increasing number of episodes in movements 1 to 3,
that is, four, five and eleven episodes respectively,
causes the increasingly static music to last progressively longer and longer.
All of this changes in the final movement, the "Quodlibet," which is the shortest
of the four. Here, the tempo more than quadruples,
the rhythmic values become shorter, more dance-like, and the dynamics
suddenly turn "forte." The resulting expression has often
been described as "emotionless". Cage asks the
performers to underscore this quality in the music
by playing with no vibrato and no weight on the bow. Let's listen to
the canonic imitation in the 3rd movement,
"Nearly Stationary," where the music achieves its greatest stillness.
[MUSIC]