The rhythmic infinity series

The rhythmic infinity series is really already there for the taking: our ordinary proportional rhythmic system with its semibreves, minims, crotchets, etc., is both infinite (in principle) and hierarchical; but at the same time it is almost too readily available and does not match the fascinating patterns and wave-like movements of the infinity series. Nor does it satisfy Nørgård's fondness for the expressive potential of periodic rhythmic structures. Per Nørgård found his solution in the golden section, which already had a long history as a formal musical principle, but which must be said to be new when used for creating rhythmic detail.

The golden section
The golden section is the section that divides a line in two, such that the smallest segment has the same ratio to the largest as the largest segment to the whole line.

Converted to approximate whole numbers this corresponds to, for example, 5:8 = 8:13, which again corresponds to the so-called Fibonacci sequence.
Throughout the centuries the golden section has been praised for its harmony and beauty: in the Middle Ages it was even known as the divina proporzione, since (as with the Trinity) three entities were required to explain it. It has been used innumerable times in art and architecture and is also found in a wide variety of forms in nature.

Below may be seen a line of 144 units divided on golden principles into two parts, consisting of 55 and 89 units respectively. Each of the line segments thus created is again subdivided in golden proportions, and this subdivision can continue indefinitely. It is evident from the most subdivided line that Per Nørgård alternately places the smallest and the largest element in the golden section first. This results in a proportional series that alternately expands and contracts in a gently undulating form:

    3 - 5 - 8 - 5 - 8 - 13 - 8 - 5 - 8 - 13 - 21 - 13 - 8 - 13 - 8 - 5



It is precisely this gently undulating pattern that is intended when the golden proportions are converted into relative musical durations (i.e., rhythms). Instead of subdividing the value of a note by the power of two (1 semibreve = 2 minims = 4 crotchets, etc.), it is subdivided into golden proportions. If a note is divided into two in this way the resulting notes will have the proportions 3:5, which can easily be expressed in our normal notation system (dotted crotchet + quaver tied with a minim). If a note is subdivided into four, the resulting notes will have the proportions 3:5:8:5, which is considerably more complex in standard notation (21 units in a 16 semiquaver space)


The harmonic infinity system

For the harmonic aspect, too, Per Nørgård discovered a system that met the requirements of infinity and hierarchical structure, that is, the natural overtone (harmonic) series. (Note that Per Nørgård emphasises precisely what is 'natural', i.e., non-artificial. Just as the golden section can be observed everywhere in nature, the melodic infinity series was not invented but discovered, as Nørgård likes to put it).


Overtones

In principle, the harmonic series is infinite; that it is also hierarchical is evident from the example given below. This harmonic series contains new harmonic series on each of its partials. By selecting every third tone from partial 3, i.e., the fifth (3, 6, 9 ...), each fifth tone from partial 5, i.e., the third (5, 10, 15 ...) and each seventh tone from partial 7, i.e., the seventh (7, 14, 21 ...) new infinite harmonic series arise.
The downward openness of the system - in principle going beyond the limits of what can be heard as notes - can be seen from the fact that the fundamental in the example can be perceived as an upper partial in several harmonic series with even lower fundamentals. Each note in this overtone hierarchy thus has several reference points. Tone 15 in the example, for instance, is - in addition to being the fifteenth partial of G1 - the fifth partial of D3 and the third partial of B3.


'Undertones'

With his usual interest in polarities Per Nørgård does not only see the hierarchy of partials as an ascending phenomenon. He operates just as much with its mirror image in a 'subharmonic' or 'undertone' spectrum, in which the ratios of the harmonic series, 1:2:3:4:5 etc., are replaced by their opposites, 1:1/2:1/3:1/4:1/5, etc. This is justified by the acoustic phenomenon that 'differential tones' arise when two notes sound together, and as the more these notes approach each other a spectrum is formed which is a mirror-image of the overtone spectrum: the so-called 'subharmonic' spectrum.
Per Nørgård thus chooses to view the harmonic and subharmonic spectra as the equivalent in harmony of the melodic infinity series, but these spectra are in principle infinite, and include all notes when one moves a certain way up the series. It is clear, however, that Per Nørgård was attracted to the major and minor feel that characterises the harmonic and subharmonic series as long as one does not move beyond the tenth partial. In practice, in fact, he rarely moves beyond the range where there is a clear distinction between major and minor, and in composition he often exploits the polarity between these ancient contrasts, which are now revealed, with the association of minor with the subharmonic spectrum, as true opposites. In the 3rd Symphony, for example, the conclusion of the first movement takes place an an ambience of major, laced with an abundance of overtones. The harmonic spectrum on the note C vanishes into thin air, only to float down again at the beginning of the second movement, this time in the form of subharmonics of a high C with a strong feeling of minor.

See moreover the article, Overtones and undertones.