5.3.3 The duration of the various groups

Einschub 1. Violin solo, and the continuous use of accelerando and ritenuto. Einschub 2. The percussionists of the three orchestras. Einschub 3. The three big bands, plus revolving chords in the three brass groups.

5.3.3 THE DURATION OF THE VARIOUS GROUPS Gruppen starts with a moment in g, set at 10 half notes. Basically, everything should then happen automatically, but Stockhausen makes a number of interventions.

g group = 10 half notes MM = 120 e group, relation to g is 5:4 = 8 half notes MM = 95 a group, relation to e is 3:4, which should be 8/3 x 4 = 10.6 half notes MM = 127. 142

This does not work because then all groups would have more or less the same duration. Therefore: e - a = 3:4. Seen from a , e is 3 half notes. The a group therefore starts after 3 half notes of the e group and lasts 4 half notes. And so on: a -f = 6:5, but f is one octave lower, which makes this relation 12:5. So the f group starts after 12 half notes in the a group and lasts 5 whole notes. f-e = 12:11, but e is one octave higher, which makes this relation 6:11. The e group starts after 6 whole notes in the f group and lasts 11 half notes. e- f = 7:8, but f is two octaves lower and becomes 7:2. The f group starts after 7 half notes in the e group and lasts 2 brevis: f - c = 9:13. The c group starts after 9 brevis in the f group and lasts 13 whole notes. This result does not really work because of the length of the f group.

142However, the relation g-a is now 10 half notes in ca 5 sec = = ± 75 = 120: 4 half notes in MM =127. This is a change from the pitch structure in which the vibration relations remain the same. For his theory, Stockhausen solves this by speaking of the mutual relation and accepting the discrepancy with the total relation. He mentions ambiguation: each member of a row relates to both its left-hand and right-hand neighbour. When these neighbours themselves would also have a relation, we could speak of ambiguation.