5.3.2 Pitch to group

5.3.2 PITCH TO GROUP So, a single vibration produces a pitch when it is equal or higher than 16 Hz. The next step is to construct a twelve-tone row. This twelve-tone row is just a way of notating a series of vibration relations. To make sure that all possible chromatic vibration relations are present, the row is an Allintervalreihe. The pitches are not used as such in the composition itself. All twelve of them are given a metronome number that is derived from the chromatic metronome number row.

Stockhausen starts with the idea that each tone of the twelve-tone row represents a moment as developed in his Klavierstücke. Now, we come to an awkward moment in the nomenclature. Since Stockhausen introduced the term ‘Momentform’ I have used this term so far. However, in the literature, moment in the context of Gruppen is usually named ‘group’ and therefore I will use this term in the further discussion of Gruppen.

The pitch determines the metronome number of that group, while the relation between the intervals of the twelve-tone row determines the length of that group.

Let us concentrate on the proportions of the group. Here, Stockhausen takes the vibration relations between the intervals as found in the overtone series. He makes use of the differences that occur when the series is notated from the 1st to the 17th harmonic. The major 2nd appears as 9:10/8:9/10:9/8:7 The minor 3rd appears as 6:5/7:6 The major 3rd appears as 5:4/13:16 The next step is to write down the all-interval row in the twelve positions, to establish the mutual vibration relations together with the metronome number that was assigned to each pitch. Also, the beat unit that enables Stockhausen to use the twelve-tone row in multiple octaves must be established. The g that starts the piece has the half note as a beat unit (there is no particular reason for this, it is simply the initial value). Consequently, a g1 has a beat unit of a quarter note, the G of a whole note. The entire time structure of Gruppen now looks like as shown in Ex. 38 on colour page XIV. Within this structure we also see the word Einschub (incision) three times. These three moments are composed freely; Stockhausen uses them to use his orchestra, which he splits in three, in a special way.