Is Per Nørgård's infinity series unique?


By Jørgen Mortensen

Nørgård himself has said that he discovered rather than invented the infinity series. There is a sense in which it must always have existed, but apparently no one else has ever unearthed it, and certainly no mathematician or composer.

A database containing various number series is to be found on the Internet: Sloane’s on-line Encyclopedia of Integer Sequences. The mathematician, N. J. Sloane, a member of AT & T Labs Research, edits the database. Nørgård's infinity series is included, and according to Sloane no one else has discovered it. However, the special version of the infinity series containing only two values is to be found under the name of the Thue-Morse sequence. It is accompanied by the following explanation (for those who can understand it!): 'Follow a(0),..,a(2^k - 1) by its complement'.

The special characteristics of the infinity series
One may therefore ask the question whether there exist other series of numbers with the same characteristics as Nørgård's infinity series. Put briefly, once can say that the special characteristics of the series can be reduced to two conditions (after the first elements have been determined as 0 and +1):
  • every other element (from the first) produces the series inverted; and,
  • every other element (from the second) produces the series non-inverted, 'transposed'.

These are sufficient conditions for the creation of the series. From the first if follows that every fourth note produces the series non-inverted, every eighth, the series inverted, every sixteenth, the series non-inverted, and so on. From the second condition it follows that the next new note in the series will also form the series, further transposed and on another 'wavelength' - in the last analysis, then revealing the special characteristic that each new note re-creates the series. (In fact, this is yet another way of constructing the series - the sixth!).

I have observed the attempts of mathematicians to create infinity series that are 'just as interesting' as Nørgård's, but have noted that they overlook this very special quality: that every new note - well, in fact, every note, period - re-creates the series.

It would appear that other series which also possess this quality are to be found, for example, in irrational integers, but mathematicians have not yet apparently become aware of them.

So strangely unique are the qualities of the infinity series!

Non-inversion/
inversion
However, it is relatively easy to construct a single series with roughly the same qualities: if we change one of the initial conditions in such a way that every other note re-creates the series non-inverted instead of inverted, the following series appears:
This series is mentioned in the section on Construction with binary numbers. That Nørgård has never found it interesting in musical terms is very understandable, as it seems monotonous and mechanical.

The conclusion is, then, that Nørgård's infinity series is actually quite unique.