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John Cage and percussion instruments – Part 2: Square Root Formula

Square root formulaIn my last post we saw how the interest towards noise as musical material encouraged John Cage to write pieces for percussion instruments.

Alright. But how to write noise on staff paper? On which basis should you build up such a music? Rhythm, harmony, pitches?

Today, we’ll talk about the “square root formula”, John Cage’s answer to these questions in his first production. A structural – rhythmic principle he built the majority of his first pieces for percussion ensemble on.

What the square root formula is and how Cage got to this will be the topics of this post.



In the article entitled “For New Sounds” on Modern Music Magazine, Volume XIX, p. 245,  you can understand how his interest towards orchestrating for percussion instruments induced Cage to develop new attitudes towards music structure.

In this article, Cage took into account the fact that a composer who writes for percussion instruments has surely to face material that doesn’t cope well with traditional scales and harmonies. Given this and the fact that the character of duration is the only one that can be easily controlled and put in relation with the different other elements, it’s logic that the whole organization system has to be centered on rhythm.

Towards the end of 1930s, Cage started exploring what he called “micro-macrocosmic rhythmic structure”, which Lou Harrison called instead “square-root formula“. Cage used this formula as the structural basis of much of his musical production, despite some important differences.

The main idea of the square root formula is this:

  • the piece is divided in n parts;
  • every part n is divided into other n parts (where, obviously, n=n);
  • the series of these n parts is established by a proportion that is equal both for the micro and for macrostructure. In other words, the macrostructure is an enlarged projection of the microstructure, or else, the microstructure is a summarized projection of the macrostructure.

Did you understand any of this? I guess not. I’ll use an example to explain myself better. Take “First Construction in Metal”, a composition for percussion instruments written in 1939 by John Cage. In this, the square root formula has an application that we could define as educational.

The piece is divided into sixteen parts, and every part is divided into sixteen measures. Both parts and measures follow the proportion (square root formula) 4/3/2/3/4.

Regarding the macrostructure, this means that the first four parts have an exposition function, whereas the following (3/2/3/4) represent the piece development.

Regarding the microstructure, this is clearly determined by changing the instrument combination: a combination for four measures, then another one for the following three, another one for the following two, another one for the following three and another one for the last four.

First Construction in Metal - square root formula

Proportional division 4/3/2/3/4 in the first 16 measures of First Construction

All the compositions that Cage wrote between 1939 and 1952 were written drawing inspiration from a similar structural principle, despite the square root formula is not always applied in the same way. Among these pieces for percussion instruments are those like Constructions and Living Room Music. 

But where did Cage get this idea of the square root formula from? There are 3 main factors.




TablaFirst of all, Cage meets Henry Cowell in 1932 and studies oriental music with him at the New School for Social Research in New York. The composer studies the structure of rhythmic cycles used in the East Countries, like the Indian Tala and Richard Kostelanetz, in the book “Conversing with Cage”, even writes that “ this [rhythmic-composition approach of Cage] is analogue to that of the Indian Tala, but it has western characteristics of the beginning and of the end”.
Beside its rhythmic-composition structure, Indian music, and especially that from North of India, is also central due to the large use that Cage makes of irregular groups, which are many in pieces like “Third Construction”.



In 1937 Cage starts his productive work on some pieces for dancing. He joins a modern dancing company at U.C.L.A. (University of California, Los Angeles) as accompanist and composer. In the same year, he moves to Seattle as composer-accompanist of the dancer Bonnie Bird at the Cornish School. 

Bird wanted to create a perfect match between dance and music, where both these arts would be freed from formalist principles. Verna Arvey writes in her “Choreographic Music for the Dance”:

“Starting by the principle that a collaboration, when too tight, between dancer and composer can lead to a conflict between personalities and to a series of little detrimental compromises both for dance and music, Miss Bird usually gives the composer a written copy of her choreographic idea… later she gives [to the composer] some indications of time so that the composer can complete the music according to his own preferences and far from her”.

As this citation suggests, Cage’s interest for modern dance surely made him think about a particular division of the rhythm of a piece, such as the square root formula.



As anticipated in the first post regarding the concept of “noise”, between 1934 and 1937 Cage studies musical analysis and counterpoint with Schoenberg at U.C.L.A. Cage himself says that Schoenberg gave a structural function to the tonality and this induced him to find some other mean suitable to percussion compositions. 

In an interview that Cage gives to Michael Williams (1988), the composer explains that in a tonal structure, sounds cannot be themselves (which means sounds do not respond to any harmonic “law”) because they follow the same rules of the harmonic structure itself. Instead, in a solely rhythmic structure there is no low like harmony and sounds are linked only by means of their duration in time. 

Here the need to organize this rhythmic structure in a well set process, a need that ends up with the invention of the square root formula.



 The attempt to write music open to any sort of noise made Cage interested in percussion instruments and convinced him to organize percussion ensembles, compose music for percussion and encourage other composers to do the same. His first works, as well as the music he initially wrote for percussion, have only been a starting point in the evolution of one of the most genial minds of the 20th century.

His new way of composing often gave little consideration to the conventional principle of tonal music, and the “square root formula” is an example of a specific compositional method created for percussion. 

His work with chance operations looks like a natural development of his composition technique. William Brooks (1982), suggested that, for Cage, the chance was simply another way to extend his determination in accepting elements which are normally refused by the classic composition procedure in the same way that rhythmic structures like the square root formula accepted the use of noise.

Cage himself said in an interview to Stuart Smith in 1983 that sound variants of percussion instruments and the unpredictable effects of prepared piano, lead him to renounce to the concept of “intention”. For example, for many of his pieces for percussion (as in Living Room Music and Third Construction) there are choices, especially instrumental choices, that performers should make and which modify the final result.

 This basically gives his pieces neverending execution possibilities, even if by analysing them we can realize that those pieces are composed following projects finely created and designed and that they require particular necessary features.

Other works for percussion and electronic instruments like “Imaginary Landscapes” (1939-1942) have been the basis of forthcoming electronic music. 

In a later interview of 1983, Cage reaffirmed his interest in the percussion field: 

I still believe in what I wrote in 1939. Percussion music is revolution. New music: new society.  I don’t think, as some seem to be thinking, that the percussion should become like the other sections of the orchestra, more expressive in their terms. I believe that the rest of the orchestra should become as noisy, poverty-stricken, and unemployed as the percussion section (or at least grant its acceptability in musical society). I do not mean anything hierarchical. I just mean accepting the fact that noises are sounds and that music is made with sounds, not just “musical” sounds.


Question: can you think about a piece by John Cage which uses the square root formula? Leave a comment.

Part 1: Noise