Concerto in due tempi

for piano and orchestra

By Leif Thomsen

Composed 1994-95

First performance 5.5.1996, on the radio, with Per Salo and the Danish Radio Symphony Orchestra, conducted by Leif Segerstam, on the occasion of the presentation of Léonie Sonning's Music Award.

Length 30 min.

Due tempi
Merging
Opposites merge
Clear-cut motifs
Pulse and carpet beater, foreground and background
Tritone
Cadence
Commentator

Due tempi

Imagine that you are watching two films at the same time. The films are not very different from each other. The players are the same and the plots are comparable. But... the two films are being played at different speeds: whilst the characters in one of the films stroll calmly from one end of the room to the other, those in the second film walk in circles and take unforeseen detours - though in the end they arrive at the other end of the room at the same time as the characters in the first film. The action, the plot, is the same, but yet not the same.

Imagine now that these two films have been copied onto the same track in soft focus. Sometimes the first film is dominant, sometimes the second, and sometimes they just shade each other out. Moreover, these shifts of perspective are unpredictable. Are you watching one film or two?

A film version of the idea behind Concerto in due tempi would be something along these lines - if indeed a musical idea could be expressed on film, or in words, and if indeed we are talking about an idea expressed in music, for this is not what the concert-goer experiences. First and foremost, the concerto is a vivid, exciting story told in music.

Merging

What is the largest possible divergence in tempo between two constant tempi? Double or half tempo? Triple tempo, or perhaps one and a half times faster or slower? If one listens to all these tempo proportions at the same time, somewhere or other merging will occur, more or less often depending on the speed of the piece. Take a look, for example, at double tempo:

Merging is a phenomenon we experience when a particular tempo is multiplied by a whole number, but even multiplication by a fraction will at some point or other lead to a merging of parts. In other words, to avoid merging one has to make use of surd (irrational) numbers.

- Opposites merge

There are various kinds of irrational number, one of which is the golden section, familiar from many of Nørgård's compositions. One of the best-known irrational numbers is , and this is the number used by Nørgård in this piano concerto. Two rhythmic layers removed from each other to the power of will never merge. They will thus create maximum rhythmical complexity from a very simple starting point - a simple formula, if you like.

At the same time, this idea offers another and very tempting perspective. Apart from the greatest possible divergence between two rhythmic layers, the proportions also allow for their merging, and in such a manner that the way is opened for an infinity series approach. This is illustrated very simply in the following score sample, in which the two-note series is written in crotchets, and then in longer notes in each subsequent notation:

See also the sketch drawn by Per Nørgård.

This sample shows a two-note infinity series with a regular pulse magnified each time to the power of - the notation in the 'uneven' layers is approximate. If you take every other note in the series, it will be repeated at half tempo, and so on. This means that the rhythmic relationship between tempo 1 and tempo contains itself an infinite number of times. Strictly speaking, one does not need to write out the other layers in the sample; they are already there.

Nørgård is thus composing with this idea in the back of his mind, though he does not do so dogmatically. It must be obvious that a completely exact performance of two rhythmic layers in a perfect ratio would be a very complicated affair. On account of the irrational numbers, even the rhythmic notation would be too complicated for performers to follow. For this reason, Nørgård uses approximations: the 7:5 (1.4) ratio is a pretty good approximation to (1.4142...).