Off by 1: Stravinsky Counting

While Mozart's "Off by 1" measuring in Figaro was certainly inspired by the playful dramatic situation of the opera, in Stravinsky this kind of counting or measuring is a recurring technique throughout his entire work. The underlying unity of thinking within the usual  tripartite division of his output (Russian/Neoclassical/12-tone) is not tonal but rhythmic. The fundamental technique is the grid, that is the superimposition of variable rhythmic patterns on an unchanging or modulating grid. An extremely clear example of this is the Bransle Gay from Agon. In this remarkable little piece, composed when Stravinsky was in his mid 70s, the grid is expressed by the unchanging rhythm of the castenet. Its two 16th, two 8th pattern is repeated 24 times creating a "floor plan" of 144 16ths. (Incidentally there is a missing 3/8 TS in the last bar.) While this  ostinato is being played, five solo instruments play music in changing meters which sometimes agrees with the castenet barring but more often does not. The durational division of the piece in its A/B/A form is, like the Mozart, a playful demonstration through "off by 1"  counting of the oppostion of regularity with irregularity.

Here is the durational form of the piece, measured in 16th notes:

A    (Ms. 310-319)   (  ) =silence

Fl/Bn/Hp/     (6)   24    (6)   24 = 2 "mobile" variations a/b/c/d (Ms. 311-314) and c/d/a/b (Ms. 316-319) These measures of 5/16 and 7/16 are -1 and +1 in relationship to Castenet 3/8
Cast/               30              30             Total Length=60

A1  (Ms. 332-335) I skip to the return because it sets in motion the "off by 1" measuring of the rest of the piece.

Fl/Bn/Hp/     (1)   16   (6) = 23  Two things of note happen here. The castenet solo is at the end and there are only three elements in the variation, variations in themselves of a/b/c (no d)
Cast/                     23 

The audible return of A is different than the notational one as Stravinsky does not begin the 7/16 on the downbeat. The "off by 1" is in itself "off by 1"!  
Total  Notated Length=23
Total  Audible Length=22

Now we can return toand since we know the length of A and A1  combined, 60+23=83, we know the length of B. It is 144-83 16ths or 59 16ths long. In other words, its length is that of A, "off by 1".

(Ms. 320-331) Ms. 320 belongs to B not A as each phrase thus far has always started with a castenet solo.

Fl              (6)     24     (5)     9   (2)     5   (10)  
Cl             (23)    (8)    (5)     7  (1)      9    (2)
Hp/Str      (52)     5      (3)     2          

Total Notated Length=59  It should be noted however, that as Adoes not begin audibly for 1 more 16th, one could say that the
Total Audible Length=60  

These variations of the three most important lengths in the piece,624 and 60 and their respective "off by 1s", 5/7,23 and 59 embody the notational joke of the piece but this is not the true subject of the piece, only the machinery that makes the true subject of the piece evident. THIS is that the function of the regular castenet, which should frame the irregular music of the other instruments, is turned on its head. As the piece progresses, especially in B, the irregular music, being more captivating than the castenet, starts to frame the castenet so that the castenet's ostinato no longer sounds regular. "Off by 1" indeed.