Introductions, part 2: Other Fibonacci applications and accent canons

In the previous post I mentioned that the form of the introduction was based on two Fibonacci series: one a series of thirteen gestures with durations of (in terms of sixteenth notes) 377, 233, 144, 89, ... 2, 1, and the other a series of twelve gestures with durations of 1, 1, 2, 3, ... 144. The first series ends with only one one-sixteenth-note gesture, and what would be the thirteenth gesture of the second series (233) gradually morphs into a new formal section.

I used the Fibonacci series in a few other ways in the piano/percussion/electric guitar texture. As I've described in the previous post, each of the thirteen gestures in this passage is a descent to the bottom of the keyboard. As the gestures get shorter it seemed unreasonable to cover the same interval of 21 semitones from C3 to E-flat1 in each gesture, so I decided to begin gradually lower each time. Here's a table that shows the initial central pitch of each of the thirteen gestures.

Except for the first and last gestures, there is a change in transposition level every two gestures. These changes follow the Fibonacci series (technically, the negafibonacci series). I chose to use the Fibonacci series here because it gives a nice curve to the entire section that mirrors somewhat the descending curve found in each gesture. I did compose the piano's final A0 of this passage outside of the system detailed in the previous post in order to make sure the goal of the lowest possible note was achieved.

I honestly can't remember if this was intentional (it has been a few days), but I find it interesting that the total number of gestures in this passage, 13, is a Fibonacci number, as is the number of semitones between C3 and E-flat1, 21. Not sure if it means anything...

Finally, I used Fibonacci numbers in a way that undermines the recursive nature of the series. I created a row of 81 (unless I miscounted..) Fibonacci numbers based on various simple additive patterns. I used this row to determine where to place accents in the piano, percussion, and electric guitar parts. I broke up the series using additive patterns in order to ensure asymmetry.

The row:

13, 8, 5, 3, 2, 1, 1, 8, 5, 3, 2, 1, 1, 5, 3, 2, 1, 1, 1, 2, 1, 3, 2, 1, 5, 3, 2, 1, 8, 5, 3, 2, 1, 13, 8, 5, 3, 2, 1, 2, 3, 2, 5, 3, 2, 8, 5, 3, 2, 13, 8, 5, 3, 2, 3, 5, 3, 8, 5, 3, 13, 8, 5, 3, 5, 8, 5, 13, 8, 5, 8, 13, 8, 13

Here is the row broken up in sections to show the various patterns.

Each part follows this row, either forward or backward depending on the part, and when it gets to the end it changes direction and traverses it again in the opposite direction. The piano begins at the beginning, the marimba at the end (retrograde), the xylophone begins on the "2" (12th line, lone 2) and moves forward, and the guitar begins on the "1" just before (end of the 11th line) and moves in retrograde. These numbers are time points on a grid of sixteenth notes. Accents fall on the first note, then again after sixteenth notes according to the row. In the end this is just a quick and effective way to ensure some variety and generate patterns in the monotonous sixteenth-note grid. I realize there is some goal-oriented motion embedded here, but this is a canon, of sorts, played by four different instruments (playing in unison or octaves, remember). The interplay among the accents on these instruments will create brief, passing motives, not the sense of direction toward a goal.

©2017 Joshua Harris