The tripartite infinity series and their variants

By Jørgen Mortensen

The tripartite series was first used in Iris (1966-68).There are many possible variations, and many different variants have actually been used in Nørgård's compositions.

The construction of a tripartite infinity series shown below is based on one particular variant, which has been chosen because it appears in the 'tone lakes'. However, this particular variant would never be used by Nørgård on its own, partly because it contains repetitions of notes, and partly because each three-note group ends on a rising minor second which sounds very monotonous. Variants of the tripartite infinity series are dealt with elsewhere.
The starting note, G, is established and the next note is placed two semitones below (F). Then the third note is added by moving one step above the previous one. This will be F sharp:                              
After the initial three-note group, the rest of the series is appears as projections. First, the interval between the first two notes (G and F) is projected. This is done first of all in one system (from G), and inverted (this will be A), then in the other system (from F), non-inverted (this will be D sharp). Finally, the third system is projected non-inverted: one whole tone below F sharp gives E:                                                                          
The order of the projections is therefore: inversion - non-inversion - non-inversion. The next interval to be projected is F - F sharp, which gives A flat E and F.
                               
The process continues in this way. The first 81 notes will look like this:                                                                                    
The tripartite series will also expand the ambit of their tonal range, slowly but surely. The notes at the extremes will remain in the middle of the whole structure all the time.

 
Hierarchical qualities
 If one selects every third note, the row becomes inverted; select every ninth note and the row will again be non-inverted. In other words, we recognise here the 'hierarchical' qualities of the split rows, though with slightly altered numerical values.

A striking graphics parallel to these series is to be found in 'the tripartite tree'. In this case, too, the construction develops through 3, 9, 27, 81, 243 and 729 elements. The basic motif, which in this case is three connecting lines, is maintained throughout.

Correspondingly, the basic motif in music, the first three notes, shows the greatest fluctuation in the middle - and this is enlarged time after time. We could easily construct 729 notes analogous to 'the tripartite tree' by establishing the first three notes of the series as nos. 0, 243 and 486, and then go on to make smaller and smaller divisions.

The tripartite series has so far been demonstrated in a chromatic scale, where each step is a semitone, but it could just as easily have been used in a diatonic scale. Alternatively, one could set each interval to a fifth interval instead of a semitone step.

Fifth intervals
Using fifth intervals, the first 9 notes of the series shown above would look like this, starting from G:
As may be seen, one soon reaches registers that are so high or so low that they can hardly be written in notation, never mind heard. One of the solutions to this problem is to set limits to the register, for example, an upper one at c’’’’ and a lower one at C. If the notes move out of the register they can be transposed 5 or even 10 octaves down or up. This principle is used when establishing the first 27 notes:                             
See also the section on variants of the tripartite series.