The 'open hierarchies' of the infinity series

By Jørgen Mortensen

The most amazing thing about Per Nørgård's infinity series is that one keeps on discovering the original row 'on another scale', that is, extended in time, at another pitch, inverted and non-inverted. The individual musical element - the individual note - is not just simply a particular number in a particular series; it appears in an endless number of different contexts.

Per Nørgård uses the term 'open hierarchy' to signify these characteristics, referring to the concept used by Arthur Koestler in Beyond Reductionism (London 1969).

However, this notion of 'hierarchy' is somewhat different from the normal use of the term to indicate a pyramid of power. Koestler saw hierarchical organisation as a condition for life, and was one of those who paved the way for an holistic view of the world. In an open hierarchy the various layers are structurally connected, but no layer is superior to another. There is no final 'top' and no final 'bottom'.

That this is also the case in Per Nørgård's infinity series can best be seen by the fact that one can rediscover the series by reducing the number of notes - by removing every 4th or 16th note - or by 'adding more notes on'.

There are two concepts from fractal geometry which are well suited to illustrate the open hierarchy: self-similarity and scale invariance. The first concept indicates that the structure is found within the structure; the second one indicates that the structure reappears on another scale.

Even the fact that now and then fragments of the original series appear within the series can be seen as an instance of self-similarity.
The series contains the series, contains...
No top,
no bottom
Two mirror halves
As the series is constructed by projection, it will be obvious that the two halves mirror each other. Therefore, if we remove notes 0, 2, 4, 6... we get the series inverted. If we remove notes 1, 3, 5, 7... we get the series non-inverted again, only transposed one step above the original - yet another instance of self-similarity.

If we remove every fourth note (0, 3, 7, 11 ... ), we get the series in its original form, and this will also be the case if we remove every 16th note, every 64th note, etc. See the sample score below:


This score sample also reveals that every time a new note appears it will form the basis of a new version of the series - transposed and at another layer in terms of tempo. We have already seen this in the case of the second note. The third note has the value -1 (F sharp). If we count from 0, we get the following values:

    pn[2] = - 1
    pn[6] = - 2
    pn[10] = 0
    pn[14] = - 3
    pn[18] = - 2

If we add 1 to all these values, we get the values 0, -1, 1, -2, -1 ..... , which is the series in inverted form.

In the same way the following values arise from the almost new note, A:

    pn[3] = 2
    pn[7] = 3
    pn[11] = 1
    pn[15] = 4
    pn[19] = 3

If we subtract 2 from each of these values, we get the series non-inverted.

The next new note, F, gives us:

    pn[6]   = - 2
    pn[14] = - 3
    pn[22] = - 1
    pn[30] = - 4
    pn[38] = - 3

If we add 2 to all the values we will have the series inverted.

On the basis of every new note - well, in fact, every note, period - the series is re-created on a new 'wavelength', non- inverted or inverted!

This is a quite remarkable characteristic!