Violin concerto

(For a detailed analytic account of this movement and the connection between the compositional technique and the composer's view of the world, see Svend Hvidtfelt Nielsen: "Virkeligheden fortæller mig altid flere historier (Reality is Always Telling Me New Tales)").

The last movement of the violin concerto (191 bars) ends and finishes with a solo cadence. These two cadences are the same length. In fact, they are identical; the only difference is that one is played forwards and the other backwards. One of the statements made by this movement is that there is a special reason for doing this.

Between these two cadences runs the orchestral movement, which in principle is one long, uninterrupted ritardando. In terms of compositional technique, it can be divided into two waves.

1. Wave 1, bars 26-93, is composed as a descent into the Super Sequence.
2. Wave 2, bars 94-170, is composed using the infinity series.

Wave 1- descent into the Super Sequence

At bar 26, the music explodes in a violent whirlwind of sound, which gradually quietens down over the following 13 bars. The ritardando flattens out and the movement retards, getting slower and slower. The music is composed as a journey down into the Super Sequence, which thus become the whirlwind that ever decreases in tempo. And as this whirlwind spins slower and slower, the soloist can identify - and play - an increasing number of the notes in the Super Sequence. The descent of the ritardando down to a tempo at which all the notes can be identified takes place in 5 stages, at each of which the melody played by the soloist divides and multiplies. Each time round, that part of the Super Sequence from which the soloist finds his or her melody moves up a minor second, and at each stage the Super Sequence takes in an increasing number of clusters. See one of the sections concerning the background of the tone lakes for more information about this.

Playing with the material

From bars 66 to 94 there is yet another plateau in the constant ritardando of the movement. At this point, Per Nørgård once more composes a section using the first couple of hundred notes of the Super Sequence. On this occasion, most of the material is placed in the hands of the soloist, who, with but light accompaniment, plays a passage composed using the first 108 notes of the Super Sequence.

Wave 2 - pulse reduction

After 'The Wild Bride' breaks into the music again at bar 94, there occurs a transformation: the theme, like a kaleidoscope, is shaken into a new pattern. The soloist divides the theme into three parts, which run along each at its own tempo, but interwoven, just like the three melodies that formed the basis of the third movement.

Each of these themes now becomes the starting point for an infinity series. Per called this version 'The Shaken Bride'.

Per Nørgård uses these three rows of the infinity series to compose the ritardando in the last section of the movement.

In more detail, the ritardando of Wave 2 consists of two forms of pulse reduction:

1st part of Wave 2:

1) This is a pulse reduction in which the three different subdivisions of the basic pulse (eight, five, three) take it in turns to form the foreground of the music, though the tempo remains unaltered. For the sake of precision, Per Nørgård has arranged the notation in such a way that the layer that is in the foreground bears the pulse, but in terms of absolute time the tempo does not change. The three layers remain the same; only the focus changes.

In this situation Per Nørgård makes use of the hierarchical qualities of the infinity series. The point is that both the quintuplet layer and the triplet layer are stressed twice - at the same tempo! The fact that they can represent two different tempi is explained by the idea that we hear them first time round as a slower layer of the infinity series. Ideally speaking, when each of the layers is repeated, the music moves along so slowly that several notes of the infinity series can be heard.

This passage ends in an oasis (bars 134-146), where the infinity series allows Per Nørgård to include a quotation from the 2nd movement. In this way the composer has been inspired to create a picture including ingredients from all four movements.

2nd part of Wave 2:

The movement continues to expand, though the steps are increasingly short. Instead of changing the rhythmic notation, Per Nørgård now emphasises the slower layers by using accent marks. The focus shifts from layer to layer at shorter and shorter intervals, until the point where it shifts so fast that the process becomes an accellerando proper.

By using this concluding 'accelerando', Nørgård contrives at the same time to suggest a reverse progression - as if the movement had done an 'about face' and was now playing in reverse time - or even a signal to the effect that the whole movement can be read backwards, as a monstrous accelerando culminating in the whirlwind at bar 26. In terms of the movement as a whole, the fact that we can imagine this situation is the very point and statement of this retrograde concluding cadence.

It turns our that this statement is significant for the further development of the Super Sequence. The idea of the double time technique inspired Nørgård to lengthen parts of the sequence in a retrograde sense (i.e. 'backwards in time').