Construction by binary numbers

By Jørgen Mortensen

It would of course be very useful to be able to predict the note occurring at any random place in the series without having to go through the processes of iteration or recursion as already described, for example, to be able to identify which note occupies the 18th position without regard to its surroundings.

When looking for a solution of this nature, it soon becomes evident that Per Nørgård's infinity series is very closely related to binary numbers, that is, the number system using only noughts and ones (0 and 1) which, of course, is used in every computer.

In the three columns below, the series of numbers from 0 to 32 first appear written in normal numbers (the decimal system). The next column shows the same numbers, but this time in the binary system. The third column shows the infinity series expressed in intervals. What is the connection here?

It would appear that in some cases one can find the value of a series by taking the sideways sum of the binary numbers, for example, in the case of no. 15 (four ones, the sum of which is 4).

If we take the first value, 0, it matches the sideways sum. The same applies to the next value, 1. The third value should really be 1, but it is -1. This is due to the fact that the first projection is inverted. We can take the last 0 in a binary number as a 'signal' of this, and thus state that in general a 0 reverses the polarity of the value.

At all events, the following procedure, which I have arrived at in a completely unscientific manner, gives the right result:

One reads the binary numbers from left to right and adds all the ones together.
Each time one passes 0 the polarity is reversed.

An example:

18 is written as 10010 in binary numbers.

    First we read the 1
    Its polarity s changed to -1 by the following 0
    Its polarity is changed to +1 by the following 0
    Adding the following 1 makes 2
    The following 0 changes the polarity of the number to -2

-2 is therefore the value of the 18th element counting from 0.

From this it also becomes clear that we arrive at new 'extreme values' at ved 0, 1, 3, 7, 15, 31, 63, 127, 255,  etc.

One possible infinity series, which Per Nørgård has never used, is not projected as non-inverted - inverted, but as non-inverted - non-inverted.
It looks like this:

0 1 1 2 1 2 2 3 1 2 2 3 2 3 3 4 1 2 2 3 2 3 3 4 2 3 3 4 3 4 4 5

It may be seen that the values in this series can be simply worked out as the sideways sum of the binary figures for the numbers in the row without regard to the noughts.

The corresponding score looks like this: