The golden section in Per Nørgård's oeuvre

Aperiodic rhythm

Per Nørgård has been greatly taken up by the use of the golden section to divide up time - into rhythmic patterns. The golden section provided a solution to the problem of an aperiodic rhythmic pattern - something that has always fascinated the composer - and a hierarchical structure.

Hierarchical structure

Instead of a completely even pulse in semibreves, one could subdivide the time on golden principles into 55 + 89 units - two numbers from the Fibonacci sequence.

A further, hierarchical development of this principle, in which the subdivision continues on golden principles, may be seen in the following table, which uses the Fibonacci numbers (note the resemblance to some of the shapes shown in the section on The Fibonacci sequence in nature):



As may be seen, the number 144 is subdivided according to the golden section into 55 and 89. Then 55 is subdivided into 21 and 34, 89 into 55 and 34. Note the general principle applied here, that when subdividing the sections one begins alternately with the shortest and the longest elements. In this way one avoids placing segments of the same length next to each other.

The subdivision continues down through the levels until we reach a proportional series that alternately expands and contracts in a softly undulating form.

3:5 and 5:3.

If 4 values are required, the proportions will be:

3:5:8:5 and 5:8:5:3

If 8 are called for, the proportions will be:

3:5:8:5:8:13:8:5 and 5:8:13:8:5:8:5:3

This principle can be applied to polyphonic movements in which all the parts contain the same number of notes and one wants to produce an aperiodic rhythmic pattern. Often in Nørgård's music there is a fundamental layer, a fifth layer, a third layer and a seventh later, corresponding to the transposition of the infinity series according to an overtone pattern.

The fundamental layer moves at an even pulse. On the next level, the fifth, the infinity series develops with the proportions 3:5 and 5:3; at the third layer with 3:5:8:5 and 5:8:5:3; at the seventh layer with 3:5:8:5:8:13:8:5 and 5:8:13:8:5:8:5:3; at the ninth layer with ... , and so on. Incidentally, it is not always easy to express these proportions using standard notation!

An example of this is to be found in Libra, where the following 'core movement' plays a major role:

The same pattern can be expressed graphically:



See moreover the analysis of Libra.

A magnificent example in six parts is the following, from Symphony No. 3, at bar 124:

See moreover the analysis of Symphony No. 3.

The following score sample demonstrates another way of writing a polyphonic movement using the Fibonacci sequence:

The fundamental layer has only one note with the value 55 (inverted row).

The fifth layer has 2 notes with the values 21 34 (non-inverted row).

The third layer has 4 notes (inverted row): 8 13 21 13.

The seventh layer has 8 notes (non-inverted row): 3 5 8 5 8 13 8 5.

These numbers can be read off directly in the left section of the figure shown above:



It may also be seen how this use of rhythms lives up to the principle of an open hierarchy: in principle, there is no 'bottom' and no 'top'; no level is more important than any other; all are parts of a golden proportion, and are themselves subdivided on golden principles, indefinitely.

Because some of these rhythms are very complicated to write in notation, Nørgård has developed a special form of rhythmic notation. In the score of Canon for organ, both standard notation and this special notation are used: