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.


I've noticed that in the past few years I've come to rely more and more on generative systems or algorithms--various devices that output specific pitches and rhythms. It's not that I don't have musical ideas spontaneously arising in my mind, it's that those ideas are usually textural, not thematic. In other words I hears sounds and shapes more often than specific melodies. Dictating a texture directly from the brain has proven a complicated challenge for me, so I've developed intermediary techniques. This is the story of the system I composed on Friday to get the right texture from the vocalists in my dissertation.

I chose the text from Samuel Beckett's

Waiting for Godot

. At the beginning of the second act Vladimir sings a short recursive song that could loop indefinitely. I see this song as a microcosm for the entire play, and think it represents the aesthetic of my dissertation piece quite nicely. Here is the text in French (the original language of the play) and in English (the playwright's native language) from Beckett's translation.

Un chien vint dans l'office
Et prit une andouillette.
Alors à coups de louche
Le chef le mit en miettes.
Les autres chiens ce voyant
Vite vite l'ensevelirent
Au pied d'une croix en bois blanc
Où le passant pouvait lire:
Un chien vint dans l'office
Et prit une andouillette...etc.
A dog came in the kitchen
And stole a piece of bread
And cook up with a ladle
And beat him until he was dead.
Then all the dogs came running
And dug the dog a tomb
And wrote upon the tombstone
For the ages of dogs to come:
A dog came in the kitchen
And stole a piece of bread...etc.

I knew I wanted this text to evolve in a specific way: beginning as vowels, gradually changing to non-sense phonemes, and finally becoming clearly understood as language at the very end of the piece. This process will probably begin around five minutes into the piece and take around 10 minutes or so to work itself out. The trick was to find a logical process by which the text could gradually evolve over that period of time.

I started with the text, then broke it down into syllables. Here's the 56 syllables in the English version:

ə dɒg kem ɪn ðə kɪt tʃən

ænd stol ə pis əv brɛd

ðɛn kʊk əp wɪθ ə le dəl

ænd bit hɪm ən tɪl hi wəz dɛd

ðɛn ɒl ðə dɑgz kem rən nɪŋ

ænd dəg ðə dɒg ə tum

ænd rot ə pɑn ðə tum ston

fɔr ðə edʒ əz əv dɒgz tu kəm

I had to manipulate the French text a little bit to get 56 syllables:

œ᷉ ʃjɛ᷉ vɛ᷉ dɑ᷉ lɔ fis

e pʁi yn ɑ᷉ du ɪ jɛt

a lɔʀ a ku də luʃ

lə ʃɛf lə mi ɑ᷉ mjɛt

le zo tʀ ʃjɛ᷉ sə vwa jɑ᷉

vi tə lɑ᷉ sɛ vɛ li ʀɑ᷉

o pi je dyn kʀwa zɑ᷉ bwa blɑ᷉

u lə pa sɑ᷉ pu vɛ li ʀ

So with each text now divided into 56 syllables and each one labelled 1-56, I could use numbers to effect an even distribution of changes over several iterations of text. I used a random sequence generator to create a the order of syllable for the first iteration of text. I created a different initial sequence for each of three soprano parts. The first two sopranos will sing "toward" the English text and the third sopranos will sing "toward" the French text. Here are the basic distributions for the first two sopranos:

Soprano 1
Soprano 1

Soprano 1

Soprano 2

In the charts above each column represents one sequence of syllables. The first column on the left contains the first


 sequence and the last column on the right contains the syllables in original,


, sequence. The changes found in each iteration of the text (i.e. each column, top to bottom) are shown, otherwise, the empty squares mean nothing has changed at that position in that sequence. I left most squares empty so that the changes will stand out.

The pattern of changes is mirrored between the two English parts above. In the first, two "correct" syllables enter the sequence on each iteration, beginning at the middle first, then expanding outward. In the second, "correct" syllables enter the sequence at the beginning and ending first, continuing toward the middle. I opted not to move directly from center to ends and

vice versa

 because I felt that with 26 sequences intelligible text patterns would become apparent too early. In other words, if the soprano 1 began to make sense with a string of intelligible words by the third or fourth iteration, that string would simply get longer each time via a simple additive process that would get extremely boring. By reiterating the pattern four times nothing becomes intelligible as language until the final few iterations. There are persistent gaps in the middle.

Soprano 3

For the French text in the third soprano part I doubled up on changes and combined both patterns shown above.

I liked the pacing of this part and began to wonder how to deal with the difference with the pacing in the other two parts. They take 26 iterations to arrive at the original text, while this one only takes 16 (there are some minor errors in the Soprano 3 chart). Since these parts will probably be proportionally notated I didn't need to figure out how to fit 16 into 26--I could just pace it more intuitively.

But then I started wondering if there was a way to speed up the pacing of the other two parts. I have used this method of reiteration with small changes in the past. The third movement of


 for string quartet is basically one 10-bar melody that takes 17 iterations to moves from 10 bars of one single pitch to the "original" 10-bar melody at the very end. I like this idea of revealing the original idea at the end. There are pitfalls, the biggest of which is pacing--it can be boring without tweaking along the way. The other major pitfall is creating a sense of linearity. In other words, the sense that the music is progressing directly along a straight path. Without getting into more detail, it's the difference between a straight line and a curve. The music should move along a curve, not a straight line. Straight lines arise from one-to-one relationships. I try to build more complexity into my systems so the outcome doesn't feel too predictable.

So, here are the revisions I made to the first two soprano parts. I've overlapped the iterations of the patterns. I think this will improving on the steady (plodding) pacing and shortening the process and also by increasing the number of changes in each iteration.

Soprano 1, revised

Soprano 2, revised

The process at this point is simply filling in the empty squares with whatever is left of the empty square:

Soprano 1, mid-process

Soprano 1, complete

Soprano 2, mid-process

Soprano 2, complete

Soprano 3, mid- process

Soprano 3, complete

Finally, I used find and replace in Excel to fill in all the syllables in International Phonetic Alphabet symbols.

Footnote--I will begin with vowels alone and gradually add consonants in the piece.

Composing in the void

I vaguely remember Steven Johnson talking about the patterns in Feldman's Why Patterns when I was in his 20th century history class back in 2007. Yesterday I came across the collection of essays he edited,The New York Schools of Music and Visual Arts, and was happily surprised to see that the volume ends with his essay "Jasper Johns and Morton Feldman: What Patterns?" I haven't finished it yet, but the description of patterns in Johns's cross-hatch painting Usuyuki stopped me in my tracks. I'll paraphrase Johnson's description (using a different painting from the same series) and show a variation on an example in his book.

Johns,  Usuyuki  (1979)

Johns, Usuyuki (1979)

Usuyuki is good example of stasis through motion. It appears kind of all-over, random, on the surface, but on a larger level it moves downward as it moves left to right (I assume you can see the division of the work into 3 panels, and each panel into 9 self-contained blocks). What I find especially interesting is that the system (it's not at all random) is not restricted to the visible portion of the work. Or a better way of putting it is the system includes content that will never be seen by the viewer. Of course blocks A-F will eventually show up in the second and third panels, but they actually exist as part of the concept in the first panel where they are not seen.

Last spring I wrote a piece called Don't cross the streams for solo horn and computer music that is based on the idea of multiple "streams" happening simultaneously despite the fact that the horn meanders among these, and only one "stream" is heard at any point in time.

I'm reminded of a criticism of Ligeti's music along the lines of Why write such intricate and systematic canons when they will never be audible in the texture. So why create something that will never be seen or heard? This gets me to the Void, an idea that deeply informs my aesthetic.

The void as I understand it comes primarily from Seyyed Hossein Nasr's writings on Sufism, or Islamic mysticism. He summarized it this way: "…[T]he void, or that which is empty of things becomes a trace and an echo of God in the created order, for through its very negation of ‘things’ it points to that which is above and beyond all things. The void, therefore, is the symbol of both the transcendence of God and His presence in all things" (Islamic Art and Spirituality, 186). This is one explanation of aniconism in Islamic art, which moved Islamic art toward abstractions centuries before Western modernism. But, the concept of the trace and echo resonate with me. Negation, inversion, etc. are profound ways of pointing directly at something. Those inaudible or invisible parts of a system (the "created order") are echoed by the audible and visible parts. They are often crucial to a complete understanding of a work of music or art.

©2017 Joshua Harris