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]