Golden Section in Webern's Canata, Op. 31 #1
by John D. Miles

Positive and negative golden sections generally articulate important structural divisions in the piece.  Note series of positive golden sections which begin at overall work’s "negative" golden section at m. 18.  The section where the time unit changes (m. 26-38) begins at a negative golden section point and ends at a different positive golden section point.  This is easier to look at than to write about it, so take your time and try to understand this diagram, because what Webern does is rather ingenious.  I was not surprised to learn that the segment which begins at the negative golden section point, is at the "deficiency," or "ausfall," which is the structural mid-way point for the Row Structure (see Appendix #1 in my paper).  Webern’s use of retrograde in the work  reflects his use of "retrograde" golden sections.  Nothing outstanding occurs at the work’s positive golden section, by the way.