Accellerando/rallentando

Brief explanations:
The first three sections below only refer to accelerando. However, the techniques can all be applied to ritardando, though obviously in reverse.

1.

Accelerando controlled by the musician using periodic accentuations

    The illusion of an endless accelerando can be created by a single musician, for example by accentuating every third note in a repeated, accelerating movement. There comes a time when only the accentuated notes can be heard - the original notes have become too fast to be either played or perceived. This procedure can be repeated ad libitum.

2.

Accelerando/ritardando controlled by the conductor

    When using an orchestra, Per Nørgård creates the illusion of endless accelerando by letting the conductor accelerate the music up to double tempo, after which he beats half time and then repeats the procedure, whilst the musicians interpret, for example, their quavers as semiquavers and continue the movement in this way.

    This process can be extended further, in that the conductor can accelerate up to more than double tempo before going over to half time, which will thus be slightly faster than the original beat. This process can of course be repeated, thus producing what one could call an accelerando of accelerandi.

    Sometimes Nørgård leaves the accelerando to the discretion of the conductor, at other times he notes the precise metronome figure for each measure.

3.

Acceleration content

    If an accelerando process is to be continued indefinitely, new, slower levels have to be constantly added which can then be accelerated. In order to preserve the illusion 'that it remains the same', the levels have to be interrelated, and therefore Per Nørgård often uses infinity series for acceleration. By doing so, he is able to produce faster tempi by fitting in new notes between the existing notes following the principles of the infinity series.

    In the case of an orchestra, several different tempi can exist simultaneously. They are marked as polyrhythms (triplets, quintets, etc.). Nørgård calls the acceleration of such levels at different tempi 'multi-dimensional accelerando'.

4.

Ritardando as the continual appearance of more material

    Nørgård has also used the properties of the infinity series to produce a qualitatively different form of accelerando and ritardando, based on the idea of a fixed number of notes determined in advance, as in a very long passacaglia theme. If the structure one explores is an infinity series, this 'theme' will, in principle, be endless.

    If the whole theme is played, the given tempo is played in its slowest form. Acceleration then takes place by gradually leaving out more notes. Conversely, the tempo can be retarded back to the theme by at first only playing the first and the middle notes, and then gradually expanding until the whole theme is present. In order to complete the illusion of a tempo shift, the movement can be carried out in combination with the technique described in section 2).

5.

Formal ritardando

    Using a very long 'passacaglia' as a basis - a 108-tone lake, for example - Nørgård is able to suggest a formal ritardando. Different sections can build on increasingly larger (or smaller) selections from this tone lake. For example, within a 108-tone lake, there are two possible slower versions, a 36-tone lake and a 12-tone lake. (Why is this so? See the section on Tone lakes)