Acceleration content

A ) Changing identity

Let us take another look at the example of what we have called 'accelerando controlled by the musicians':
The basic principle behind the slower accentuations of the rising movement is that 'the same effect' in some sense or other is recreated at a slower tempo, so that the illusion of an endless accelerando is maintained. It is therefore important that this should not just be a question of a new phrase being slowly introduced, and equally important that the faster and slower tempos should fall on the same notes. In other words, an opening phrase must never involve a double note. The slower tempo grows out of the faster.

Now, of course, in the example shown here the new figure is not completely identical with the old, as the slower level reverses the direction of movement. And yet it is not something completely new; it is still a movement along the scale. At the same time, there is no reason why the slower accentuation should not be made to recreate the rising line. It is just a question of accentuating every fifth note instead of every third.

If both versions are played, one soon realises why Nørgård chose to write the music in this way. The reproduction of exactly the same movement at a slower tempo just keeps the music idling along without a spark of excitement. It lacks the fresh input and the dynamism created by the shift in direction.

This realisation points to a particular compositional technique.

B) The infinity series

In its original form, the infinity series is based, firstly, on a straightforward principle that involves extending a given interval by first marking it out inverted and then again uninverted, and secondly, on a hierarchical structure. That is, it re-creates itself at slower tempi, as slow 'accentuations' which can be made to accelerate continuously. At the same time, the series demonstrates a development, which parallel to the re-creation of itself at slower tempi also introduces new material in an 'unforeseeable' manner. There is a freshness about this self-renewal, and yet an identity in its periodicity.

C) Transparent structure -

At the end of the 1980s and the beginning of the 90s, Per Nørgård composed three sole concertos: for cello, viola and violin, as well as a symphony, his fifth. In all these works, Per Nørgård uses a transparent structure, in which up to three different subdivisions of the bar develop intermingled with each other without ever becoming confused - see the score sample below:

naccel03.gif (3369 bytes)

The triplet subdivisions move from three beats to triple beats. The quintuplet subdivisions move from one beat to single beats and in between; on the unaccentuated sixteenths, which are not influenced by the two other levels, there is room for eight eighths. In this way the bar contains levels with 3 - 5 - 8 events respectively. Note that the three figures are related as a golden proportion [see the later link to the golden section theory], which is no coincidence. In other works, Per Nørgård has delved very deeply into just this relationship.

- multi-dimensionality

When one accelerates (or retards) within such a structure, something special happens, and in at least two ways.

    1. Phrase length and accelerando sequences
    If one imagines that each layer phrases in four-note groups, what will happen is that, at the point when the tempo is doubled and the shift to half time occurs, one or two layers will unavoidably be heard as being in the beginning, the middle or the end of a phrasing unit. This phenomenon will be even more clear if one imagines that the acceleration does not continue, but stops at the transition to half time.

    For example, in the case of a period with, let us say, 6 bars from the initial tempo to double tempo, only the quaver level with its (8x6=) 48 notes will have finished a four-note phrase at the point of transition to half time. At this point, both the triplet and quintuplet layers, with their 18 and 30 notes respectively, will be in the middle of an accelerating phrase, which then suddenly finishes without accelerando.
    In terms of what we experience, Per Nørgård calls this phenomenon 'multi-dimensional accelerando'.

    2.
    Another way in which multi-dimensionality can manifest itself is in the physical activity going on. The eighths will reach a tempo, faster than, for example, the triplet minims, at which they will have to shift over and reappear at a slower tempo (like the accentuations referred to in the section on accelerando controlled by the musicians).


Score samples © Edition WH