Karlheinz Stockhausen's Klavierstuck XI :

The analysis of the matrix system of serial polyphony resulted from the author's unaided study of Stockhausen's sketches. The duration proportion method of translating rhythm into pitch, however, was transmitted verbally to the author by Stockhausen in Paris, France in late October, 1980, and is not to be found in Stockhausen's sketches for Klavierstuck XI.This study is an analysis of how Karlheinz Stockhausen composed his Klavierstuck XI. A matrix system of serial polyphony is detailed with numerous examples drawn from Stockhausen's 26 pages of sketches for Klavierstuck XI: this reveals the manner in which number matrices, rhythm matrices, element matrices, etc., operated upon and/or combined with one another in order to create 36 rhythmic fragments, 19 of which were chosen to be translated into pitches for the music fragments which constitute the published score of the work.A separately bound appendix comes with the text of this study. It provides additional copies of selected examples also contained within the text: those examples referred to in the text, but not present on the same page, can be simultaneously viewed by consulting the appropriate example in this appendix. This is helpful because so many examples directly relate to one another in various ways.This study also describes the manner in which Stockhausen translated the rhythm fragments into pitch realizations. It then concludes with an analysis of the pitch realizations of the 19 music fragments, expresses with the inclusion of a variable formula which analyzes the duration proportion relationship between every pitch