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John Cage’s Third Construction – Part 2: Analysis

Foto by TEDx UND / CC BY-NC-ND

Foto by TEDx UND / CC BY-NC-ND

This is my second post on Third Construction, the piece for percussion quartet written by John Cage in 1941.

In my first post, I discussed and described the instruments that John Cage assigned to each of the four players.

Today I will analyze the sheet music for this piece, specifically the following elements:

  • timbre possibilities;
  • notation system;
  • use of irregular gruppetti;
  • structural analysis partially based on the square root formula.



Regarding timbre, it is important to verify if and how Cage intended each player to produce a specific number of sounds, whether achieved by using beaters or different execution techniques.

In First Construction (1939), in fact, Cage had determined that each player should have had 16 different sounds available, a number not randomly selected but carefully chosen according to the square root formula, that for that piece was based on a total of 16 sections. Since, as I’m about to show you, the square root formula for this piece ends up being based on the number 24, I wonder whether Cage had indeed planned on 24 sounds for each player.

When you examine the instruments for the four players, you can easily observe that this intention is not completely satisfiable in any of the parts because, if you consider the different beaters and sound effects that Cage assigns to some of the instruments, it’s fairly obvious that the number reached is higher than 24. Listed below are the sounds that can be combined from each player:

Player 1:

  • Northwest Indian wooden rattle: 1 sound
  • 5 graduated tin cans: 10 sounds (5 on the center and 5 on the rim)
  • 3 tom toms: 6 sounds (three on the center and three on the rim)
  • claves: 1 sound
  • large Chinese cymbal: 1 sound
  • maracas: 4 sounds (2 normal sounds and 2 muffled sounds)
  • teponaxtle: 2 sounds (1 high and 1 low).

TOT: 25 sounds

Player 2:

  • 3 tom toms: 12 sounds (3 on the center and 3 on the rim plus 6 other sounds because they’re played also with fingers)
  • 5 graduated tin cans: 10 sounds
  • claves: 1 sound
  • 2 cowbells: 2 sounds
  • Indo-Chinese wooden rattle: 1 sound
  • lion’s roar: 1 sound

TOT: 27 sounds

Player 3:

  • 3 tom toms: 12 sounds (3 on the center and 3 on the rim plus 6 other sounds because they’re played also with cricket callers)
  • tambourine: 1 sound
  • 5 graduated tin cans: 10 sounds
  • quijadas: 1 sound
  • claves: 1 sound
  • cricket callers: 2 sounds
  • conch shell: 1 sound

TOT: 28 sounds

Player 4:

  • tin can with tacks: 1 sound
  • 5 graduated tin cans: 10 sounds
  • claves: 1 sound
  • maracas: 2 sounds
  • 3 tom toms: 16 sounds (3 on the center and 3 on the rim plus 6 other sounds because they’re played also with fingers plus 4 because they’re played also with maracas, normal and muffled)
  • ratchet: 1 sound
  • lion’s roar: 1 sound

TOT: 32 sounds

Unlike First Construction, this composition shows no direct relationship between the number of sounds produced by each player and the square root formula.



Regarding the notation system, the work is distributed on a sheet music divided into 3 systems per page, each made of 4 staffs. Since all instruments, including the conch shell, produce an undefined sound, they’re played on a neutral key. Lines and spaces on each staff correspond to different heights or sounds for the same instruments. For example, graduated tin cans are marked on the spaces in between the lines when they are to be played in the center and on the lines when they are to be played on the side.

Tin cans

Example of notation system for graduated tin cans in Third Construction

This piece uses conventional notation for rests, indication of dynamics, and rolls. Each part specifies the types of beaters needed and their appropriate level of hardness as well as in which instances an instrument needs to be used as a beater on another.

Example of instrument used as a beater on another instrument, fourth player, measure 191-192

Example of instrument used as a beater on another instrument, fourth player, measure

In order to indicate long sounds as the ones produced by the lion’s roar or the conch shell, ties are used. The same type of notation is used for long rolls on tambourine, cymbals, or rattles.

Tied sound

Example of long sound, second player, measure 115-116

The teponaxtle played by the first player is noted on two different spaces depending on whether it has a high or low sound, whether the notes are pointing up or pointing down, and whether it is played with the right hand using a soft rubber beater, or with the left hand using a finger.

Notation for the teponaxtle, first player, measure 249-250

Notation for the teponaxtle, first player, measure 249-250

Every 24 measures, capital letters, that I used as a reference during rehearsals and performance, are used to mark important structural elements, as we will soon discover.

Time signature 2/2 is used for the whole piece. Gruppetti are marked with a conventional notation system; however, when they are between measures, they are often accompanied by notes in parenthesis, written either above or below the line, to highlight duration rather than rhythm.


Example of gruppetti between measures, second player, measure 222-223

Example of gruppetti between measures, second player, measure 222-223



Cage based the whole structure of his work on the use of rhythmic, irregular gruppetti. According to Cage himself, his intent with Third Construction was to create “rhythmic cadences”.

These “rhythmic cadences” are achieved through a variety of irregular gruppetti that are inserted at the end of every section of 24 measures, except for five sections: at the end of section VIII, XI, XIV, and XIX regular rhythms are present and at the end of section XV the long trill of the the cricket caller is inserted.

In many cases, these gruppetti are present in more than one part simultaneously, creating a rhythmic tension that is released at the beginning of the following section by a new predominant regular rhythm.

Example of “rhythmic cadence” in Third Construction, measure 24-25

Example of “rhythmic cadence” in Third Construction, measure 24-25



As far as the structure of this composition, Cage used the square root formula as in the two previous Construction pieces (for an in-depth analysis of this musical-structural technique, you can read my old post on the square root formula). However, unlike First Construction and Second Construction, which were based on a cycle of 16 measures each, Third Construction is based on a more elaborate system arranged in 24 sections each divided in 24 measures.

Furthermore, this piece doesn’t comply with a set proportion between macro-structure and micro-structure, as would be expected, because each player is divided into 24 measures differently.

  • Player 4: { 8, 2, 4, 5, 3, 2 }
  • Player 1: { 2, 8, 2, 4, 5, 3 }
  • Player 3: { 3, 2, 8, 2, 4, 5 }
  • Player 2: { 5, 3, 2, 8, 2, 4 }

John Cage himself commented on this diversified and unconventional structure in an interview conducted by Barry Michael Williams in 1988: “[In Third Construction] each part is divided in other parts differently… [but] none of these parts have the same structure… I like this independence.”

The following picture shows this micro-structural division, diversified for each player, for the 24 measures of Third Construction.

Proportional division of the first 24 measures for each of the 4 players in Third Construction

Proportional division of the first 24 measures for each of the 4 players in Third Construction

If the part for one of the players is structured differently from the other three, it is then legitimate to wonder whether the main structure is associated with one of these proportions and if so, with which one. The most plausible answer seems to indicate the fourth player: {8,2,4,5,3,2,}.

As a matter of fact, the first eight sections could represent the introduction to the piece, starting with a “piano” dynamics and with the use of the main instruments common to all four players, rattles, cans, tom-toms, and claves. The rhythmic and dynamic tension grows little by little, stopping suddenly at letter E, the sixth section, where the long trill of rattles marks a series of “themes” for the second and third players, still in a “pianissimo”.

From this point on, the rhythmic structure begins to evolve until the eighth section, letter G, where at measure 181, a “stringendo poco a poco” takes over until a “fortissimo” trill of maracas and quijadas marks the first rest of this piece. From here, the piece begins to develop with an interesting rhythmic puzzle of claves playing in a 4 measure “ritardando” before letter I, which ends with a “diminuendo” by the second player, introducing the following section of four parts.

This new section begins with the same tempo used in the beginning of the piece and, at letter J, with a solo of the teponaxtle played by the first player, accompanied on and off by the lion’s roar. The timbrical structure then evolves with the gradual addition of various sounds, such as quijadas, ratchet, and suspended cymbal.

At letter M, the theme is led by the first player on teponaxtle accompanied by the second player on claves until, without a real sense of continuity, the next section of five parts start with letter N, which connects the two sections, and continues with the subsequent ones, which starting with letter O, are characterized by an increasing rhythmic and dynamic excitement and by a tempo which tops off to a “veloce” from measure 361 on.

Starting with letter P, the conch shell is introduced. The conch shell has the important role of ending this large section and starting the next one with a fermata at measure 456, which comes to a close with a “fortissimo” at the beginning of letter S.

These last two sections include five parts in all and can be considered the coda of the piece. From letter S on, an even faster tempo introduces an even more complex rhythmic and dynamic structure, which culminates with letter U, the end of the second to the last section. Letter V and W conclude the piece with an even more excited finale achieved by an accelerando all the way to the end and with a dynamics that, starting at measure 553, stabilizes on a “fortissimo” for all four players.



With this post, I conclude the analysis of Third Construction, one of the best works by John Cage. Once the description, the hypothesis, the analysis of the sheet music, of the gruppetti, etc are over…. what is left is the music. Splendid, as smooth and easy on the listener’s ear as difficult to execute for the players.

It is a music that for some reason brings me back, when I listen to it, to the essence and the origin of the instruments I play, to the sense of the instruments themselves. The rhythm. The timbre.

I am going to end with my favorite execution of this piece that you can find on Spotify at the link below. It’s played by The New Music Consort, from the album “Pulse! Percussion Works”. However, I can guarantee you that when you listen to Third Construction live, it is a whole different experience. Enjoy!

Question: do you have any contribution to this piece you’d like to share? Any suggestions on the analysis I provided? Leave a comment.

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