An infinity series with only 4 or 2 different variants

By Jørgen Mortensen

Now and then, Nørgård limits himself to just four different values instead of the gradual expansion towards infinity. This procedure means only using the values -1, 0, 1, 2. The value of 3 is counted as -1 (by subtracting 4). The same applies to all higher values - one subtracts 4 or 8 or 12 or… . In the opposite direction one adds 4 or 8 or 12 or… . The result may be seen in the second system in the score sample below:
One can also apply a tonal limitation by limiting the number of notes to only two. In the case of all values above 1, one subtracts 2, 4, 6, or…. . In the case of all values below 0 one adds 2, 4, 6, or…. . After this arithmetic one is left with a new series which only has the values 0 and 1. This may be seen in the bottom system shown in the score sample.

This series is known under another name as the Thue-Morse series.

However simple the two-note row may appear, it has proved to be extremely fecund in composition - one finds it in such widely different works as Symphony No. 3  and  I Ching. A more detailed commentary on the two-note series is to be found in the analysis of the 2nd Movement of the choral work, Wie ein Kind, in connection with the treatment of Dream Songs. The section about music for amateurs contains an explanation of how to construct this series.