Concerto for Two Double Reeds in detail: II — Meta-Canon

I have already posted analyses of two aspects of this movement: the permuted chaconne in the lower strings, and the mathematical structure of the soloists’ meta-canon. In summary:

  • The lower strings play 13 permuted repetitions of a simple, 12-bar phrase
  • The oboe and bassoon play 12 phrases of different lengths, in different orders

In this post I will explain how the musical material of the solo parts is derived, and discuss a few other final details. You can also click here to read the full score of the movement and listen to the live recording*.

The melodic material is based on a 28-member interval row generated by rolling a die:

3 3 4 5 6 6 1
1 5 6 4 2 5 6
6 3 2 1 1 2 6
1 3 1 3 5 5 5

In writing the material for each section, I followed a simple set of rules:

  • The first pitch is always a D
  • Each new note is found by moving either up or down by the corresponding interval in the row
  • Any octave transposition is permitted
  • Hence in the oboe part, starting from d´, and with an interval of 3, the next note can be b, f´, b´, f´´, b´´or f´´´
  • Any note or succession of notes can be repeated, including sequences (eg u–v–w, u–v–w–x, u–v–w–x–y, u–v–w–x–y–z)
  • The pitch material for each section is derived separately, although certain techniques and motifs recur

The first two phrases of the oboe part illustrate these principles:

Eight-bar excerpt from the oboe part.

The first phrase uses the first 7 members of the interval row in sequence: 3 3 4 5 6 6 1. The second phrase repeats the notes of the first phrase, adding one more note, again generated using the interval row.

Another illustration can be found from a bassoon passage half-way through the movement:

Four-bar excerpt from the bassoon part

In this passaged again the material is generated from the beginning of the sequence, but the process is drawn out by the repetition of sequences of notes.

As the interval row is fairly large, it is only used in the 9, 10, 11 and 12 unit sections. In sections 1, 2, 3 and 4, only 7 members (one line) of the row are used for each section, and in 5, 6 and 7, 14 members (two lines) are used. Section 8 is silent.

The other patterns in the material are that in odd-numbered sections the oboe and bassoon parts are inversions of each other, while in the even-numbered sections they are rectus. Sections 1, 4, 6, 7, 9 and 12 are lyrical in nature, and sections 2, 3, 5, 8, 10 and 11 are more mechanistic (8 trivially so).

Here then is a brief analysis of each of the sections, following their order in the oboe part:

Section 12

The longest passage in the movement, the material in this section is lyrical, covering a wide tessitura. The passage falls loosely into two sections, the first  characterised by sustained notes which lead into arpeggio-like flourishes; the second by repeated notes leading into material which meanders around a small cell of pitches.

The material from this section recurs throughout the movement.

Section 9

This section is a reworking of the material in Section 12. It is rhythmically simpler and less discursive. Gestures from the solo material are picked up in the violins and form part of their ostinato (see below).

Section 6

This section picks up material from Sections 9 and 12 and develops it a little further.

Section 3

The first ‘mechanistic’ section in the oboe. This introduces the technique of gradually adding to a cell of pitches.

Section 2

This is the one point of true coincidence in the movement. At the beginning of the movement, Section 12 in the oboe is accompanied by Section 2 in the bassoon. At this point, this relationship is reversed.

The material is extremely simple: just a falling sequence notes. The number of beats between the notes decreases over the course of the 4 bar section.

Section 5

Section 5 is structurally fairly simple: a 2-bar phrase is repeated for two different sets of pitches. It’s then split in half, and each half is repeated independently. The opening motif is then repeated on its own.

Section 8

Section 8 is silent, primarily to give the soloists a chance to get their breath back!

Section 11

Section 11 is highly mechanistic. The excerpt of bassoon music above is taken from the beginning of this section, and illustrates the way the material is generated. Its nature is fairly relentless, and had to be reworked a couple of times to make sure the performers had enough space to breathe.

This is also one of the moments of minor coincidence in the movement, as the oboe starts this section 6 bars after the bassoon. This makes this passage one of the more obviously canonic parts of the movement (inverted of course, as 11 is an odd number), and the driving rhythms of the material make this the climax of the movement.

Section 10

Section 10 is fairly simple, and similar to Section 5: a phrase is repeated and then deconstructed.

Section 7

Section 7 provides another explicitly canonic passage in this movement, as the oboe starts this section 8 bars after the bassoon. The section is divided in two, and the second half is an exact inversion of the first. Because this division happens after bar 7, and because the oboe and bassoon parts are inversions of each other, this means that the second half of the section in the bassoon is in rectus canon with the first half of the section in the oboe, with a delay of one bar.

I am particularly fond of this section because of this canonic cleverness, so here are the bassoon and oboe parts on their own to clarify the structure:

Oboe and Bassoon parts demonstrating double canon

Section 4

Section 4 takes the movement’s opening motif as its sole source of material, and leads into…

Section 1

The final coincidence of the piece, oboe and bassoon play Section 1 simultaneously. The material again uses the opening motif, so in the oboe it follows directly from Section 4. As 1 is an odd number, the parts are inversions of each other, so they open away from middle d´.

Upper Strings

I have discussed the chaconne in the lower strings, which is made of 13 repetitions of a 12-bar phrase. The material in the upper strings is made of 12 repetitions of a 13-bar phrase, and uses entirely extended techniques: tapping the strings with the wood of their bows, playing the short lengths of string beyond the bridge, tapping the wooden body of the instrument, and playing notes with Bartók or snap pizzicato.

There are a few melodic elements in this material, principally echoes of the soloists’ material in Section 9, and I was keen on the idea of building a background soundworld, elements of which could then come into focus in the right context.

This movement is by far the most complex compositionally, and probably the most challenging to listen to, but I think that, despite all its complexities, it has a lyricism that works well within the structure of the whole work.

I also have a feeling that this movement would work well rewritten for oboe, bassoon and electronics. A project for another day!

Note: if you would like to listen to a recording of this performance, please get in touch and I can send you a link.

(*Please read my note on copyright.)

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