Ritardando as the continual appearance of more material

A) The infinity series

Since the infinity series re-creates itself in such a way that, for example, notes 1,5,9,13, etc. are identical with notes 1,2,3,4,5, etc., one will always be able to move in two directions. One can choose at any time whether the notes that are heard should be perceived as 1,2,3 or as 1,5,9 (or for that matter as 1,17,33, etc.).

In accelerando, the structure of the infinity series ensures that there is always a slower level to start playing on.

Ritardando uses the opposite possibility offered by the infinity series. As the notes gradually become more and more separated, the intervening spaces are filled out with notes from a faster level.

Ritardando will thus take the form of an exploratory dive down into a structure. The slower the tempo, the more notes one will be able to hear.

B) Passacaglia structure

This structure does not need to be an infinity series. On the contrary. In fact, a series of notes repeating itself in passacaglia would perhaps be more suitable.

In the case of such a structure, a repetition of the first and middle notes would indicate such an incredibly hectic tempo that all the other notes would be impossible to hear. Progressively suppressing more of the 'passacaglia's' notes indicates a reduction in tempo that makes an increasing number of notes audible. And of course, one could also do the opposite: indicating accelerando by progressively jumping over more notes.

Both the procedures described here are to be found in the last movement of Per Nørgård's violin concerto Helle Nacht. Moreover, they are to be found combined both with a ritardando controlled by the conductor, and with no other ritardando than that contained in the structure.

C) The violin concerto Helle Nacht

In the final movement of Helle Nacht, Per Nørgård wanted to set up a ritardando of a bar's length.

He achieved this with the help of two techniques.

In bars 26-93, the ritardando is set up using a passacaglia structure, 'the Super-sequence',  which permits more and more of its notes to become audible at each repetition.

In bars 94 -170, the ritardando is established with the help of the infinity series technique.

In the ritardando from bar 26, the delving down into the 'Super-sequence' is combined with a conductor-controlled ritardando, which in the space of two bars brings the music down to half tempo, after which the tempo is doubled and the procedure is repeated.

In the ritardando from bar 94, the tempo remains stable. The retarding instrumental parts play an infinity series version of a theme from the first movement, entitled 'The Wild Bride'. It is distributed over three parts playing in a transparent structure.

The ritardando is produced simply by the fact that the various layers (triplets, quintuplets, eighths), take turns at playing in the foreground.

Within this ritardando, towards the end of the movement, Nørgård indicates an accelerando. He does this by accentuating increasingly faster levels: triplets - quintuplets - eighths - sixteenths. At the same time, by using this concluding 'accelerando', Nørgård suggests inverse time, as if the movement had 'turned back', and was now playing in reverse. In accord with this, the subsequent - concluding - solo cadence for violin is a retrograde version of the solo cadence which opens the movement up to bar 26.

See also the analysis of Helle Nacht, 4th Movement.

D) Night Symphonies, Day Breaks

An example of the opposite procedure, accelerando from many notes to few, is to be found in the sinfonietta Night Symphonies Day Breaks, in which the accelerando is connected with a conductor-controlled accelerando (as described in section 2). In this case, Nørgård does not make use of the infinity series, but of his 'tone lakes', which are a kind of 'closed' infinity series with a fixed beginning and end, but which can be extended ad infinitum by putting in two new notes between the existing ones.

From bar 260 the music gradually moves into a huge accelerando, passing downwards through increasingly evident tone lakes. This movement passes from a 972-tone lake through 324, 108, 36, and finally a 12-tone lake, when the acceleration stops.