Infinity row

The infinity series is an infinite series of whole numbers that the Danish composer Per Nørgård used as the mathematical basis for compositions.

definition

The infinity series is constructed as follows: You start with a number and add a second number increased by one : ${\displaystyle a_{1}\in \mathbb {Z} }$ ${\displaystyle a_{2}}$

${\displaystyle a_{2}=a_{1}+1}$

All other terms are formed according to the following equations:

${\displaystyle a_{n}-a_{n+1}=-a_{2n-1}+a_{2n+1}=a_{2n}-a_{2n+2}}$

From this follows the rule for the calculation of the following numbers with an odd index

${\displaystyle a_{2n+1}=a_{n}-a_{n+1}+a_{2n-1}}$

as well as the rule for the calculation of the following numbers with an even index

${\displaystyle a_{2n+2}=a_{2n}-a_{n}+a_{n+1},}$

the sequence is therefore uniquely determined by the equations after the choice of . By ${\displaystyle a_{1}}$

${\displaystyle a_{2n}=a_{n}+1}$ and ${\displaystyle a_{2n+1}=-a_{n+1}}$

recursively defined sequence also fulfills the given equations and is therefore equal to the infinity series. One receives for the episode ${\displaystyle a_{1}=0}$

${\displaystyle 0,\,1,\,-1,\,2,\,1,\,0,\,-2,\,3,\,-1,\,2,\,0,\,1,\,2,\,-1,\,-3,\,\ldots }$(Follow A004718 in OEIS , index shifted by one),

the sequence for anything is obtained by adding to each term in this sequence . ${\displaystyle a_{1}}$ ${\displaystyle a_{1}}$

properties

From the recursive description it can be seen that the sequence formed from every second tone of the infinity series is the initial series transposed by one . If you start with the first link, but skip every second, you have a reversal of the starting series. If you skip two tones, you get an excerpt from the original row and so on. Logically, the same laws apply within the new series, so that this process can be continued ad infinitum - the infinity series is full of self-similarities , in mathematical terms it is a fractal . If you consider the remainders 0 or 1 of the terms of the sequence when dividing by 2, you get the Thue-Morse sequence .

In music

The infinity series (Danish "uendelighedsrækken") is used in the music of Per Nørgård, who developed it in 1959 as the basis of his music. In his works Voyage into the Golden Screen (1968) and Symphonie No. 2 (1970) it even forms the backbone of the entire composition. It does not consist of fixed tones and has no predetermined intervals such as a twelve-tone series . It can be formed using any scale (2-tone, diatonic , whole tone , chromatic , Slendro and so on). It describes the position within any pitch scale, but not absolute pitches. If the basic scale is about C major , the number 0 corresponds to c, the number 1 to d and so on. In the whole tone scale on f sharp, however, the number 0 corresponds to the f sharp, the number 1 to the g sharp and so on.

The following note example shows the first 32 tones of an infinity series in G major , with a set and g 'corresponds to 0: ${\displaystyle a_{1}=1}$

Web links

• Jørgen Mortensen: Uendelighedsrækken (description of properties of the infinity series), www.pernoergaard.dk
• Jørgen Mortensen: Infinity series, chromatic, from the start 43 '' (MP3; 302 kB) (sound sample, not composed by Per Nørgård), www.pernoergaard.dk