In the ending part, Stockhausen writes “unregelmäsiß in Einsatzabständen und Geshcwindigkeit” and. However, Stockhausen should never organize these notes in an “unregelmäsiß“ manner. When playing these groups, it is easy to notice some repeated forms. Starting from these easily recognized phrases, I have discovered a core pitch set pc5-13b[02348] pervading this ending part and the developing pattern, which is a perfect fourth sequence!
As in the figure, the appearance of a whole pitch set—let’s call it an element—is marked by a red slash, and the first two elements are circled out as examples. Later in this part, some elements are overlapped, which are also circled for reference. Each element is a permutation of the pitch set. There are at least 15 permutations appearing here, noted with Greek letters, amongst the most common ones are α(82430) and β(28043). These two are distinguished by the order of the tritone. And most of the rest permutations can be categorized into two patterns similar to α and β, respectively.
The notes in red circles are the turning points. They mostly appear between two elements as a division, however, sometimes they are hidden in an element. They strictly follow a sequence: D, A, E, B, #F, #C, #G, #D, #A, F, C, G, and another period toward the end. In most cases, the relation of these notes and neighbor element is regular, ensuring the sequence of the element, which are augemented fourth jumps or chromatic steps. Note that the perfect fourth, the augemented fourth and chromatic intervals are significantly demonstrated at the very beginning with the repeated chord [0167] and a chromatic scale.
Now let’s turn to the accented notes. There are truly amazing discoveries here. Starting from the 5/8 bar to the 2/8 bar where a full length note appears to mark a section point, the accented notes are B, E, F #A, G, D, #G, C, #F, nine in total. I find that using #F and C as the 2 and 8 tones in the pitch set to build α and β, we can obtain eight notes out of the nine accented notes, except G. And these eight notes form a pitch set pc8-25, which is Messiaen mode 6, bearing important displacement symmetry!!! The G note is even more interesting, not only because it appears right at the center. Its unique position reminds us of the notes in red circles. And it is EXACTLY the note that we need to connect the α and β elements built from the same tritone #F and C, according to the most frequent rule in the context!!!
However, something still bothering me is that there are some apparently redundant notes and missing notes. And before the very end, the bass E and two bars are not consistently explained.
Since the division based on pitch set pc5-13b[02348] works for most part at the end, I believe it’s contructive to give a hint of “Wie können an beliebiger Stelle kleinere Zäsuren variierter Länge gemacht werden”
