5.3.4 The group

Stockhausen replaces this passage and introduces a free passage (a mini Einschub) in which an accelerando is composed. This kind of relative tempo markings have no place in the Moments structure. c- b = 8:7, this becomes 2:7; c is one octave lower. The b group therefore starts after two whole notes in the c group and lasts 7 quarter notes. b -b = 11:12, this becomes 11:6; The b group therefore starts after 11 quarter notes in the b group and lasts 6 half notes (in the score it starts 2 quarter notes too soon). b-d = 5:12; The d group therefore starts after 5 half notes in the b group and lasts 12 quarter notes. d-a = 4:3; The a group therefore starts after 4 quarter notes in the d group (in the score it starts 4 quarter notes too soon) and lasts 3 half notes. a-c = 8:10; The c group therefore starts after 8 quarter notes in the a group and lasts 10 quarter notes (here the set-up goes wrong, because orchestra 1 is still playing the c group of 13 whole notes and orchestra 2 is playing the d group of 12 quarter notes, so there is no orchestra available to start the c group. (Stockhausen allows more time for the start of the c group, so that the orchestra can prepare for the change in tempo.) c -a = 10:8; The a group therefore starts after 10 quarter notes in the a group and lasts 10 quarter notes. a-b = 9:10. The b group therefore starts after 9 quarter notes in the a group and lasts 10 quarter notes. We are now at number 16 in the score, where the first Einschub starts, which lasts until number 23. The rest of the groups are defined in the same way for the entire score.

5.3.4 THE GROUP The basic structure of Gruppen has 12x12+1=145 groups, the durations of which depend on the mutual relation within the rows. The MM depends on the note that is the fundamental value of the group. Stockhausen has now arrived at the second part of his continuing.

The tone within the row must be regarded as a phase structure (see 5.3.1) with a defined number. These numbers are used to determine the duration of the groups. However, each group can be seen as a complex time range with one fundamental duration. Each group is a choice from this durational range. Just like a formant represents the intensity relation of the members of the overtone range of the tone (tone colour), likewise the durational formant is a choice from all possibilities of the

duration range. A group, expressed as a durational formant, can be formed in various ways as shown below in Ex. 39143

Ex. 39: Two examples of a developed durational formant. From: Stockhausen, Texte, band 1 (DuMont Dokumente), © Stockhausen Verlag.

143Stockhausen claims he is helped by the shapes of the mountain landscape view from his study in Davos, where he composes Gruppen.