Gradient Drawings

I started mak­ing ma­chine draw­ings dur­ing my un­der­grad. Many of my early draw­ings where born from the ex­cite­ment fo get­ting the ma­chine to work at all. Suddenly, I could draw per­fectly straight lines, re­peat ges­tures hun­dreds of times and keep draw­ing for hours at a time. There’s some­thing oddly mezmeris­ing about see­ing a fa­mil­iar draw­ing in­stru­ment (like a ball­point pen) move in an un­famil­liar way: slow, in long, straight lines and with even pres­sure.

Eventually I made an­other dis­cov­ery: When I re­peated a shape of­ten enough (or stepped back far enough), the lines dis­solved into shades of gray. I made some draw­ings that ex­per­i­mented with this, but they were never com­pletely sat­is­fy­ing.

A few months af­ter my grad­u­a­tion, an ar­chi­tect gave me a 1976 book on Iannis Xenakis, the mod­ernist com­poser. The book cov­ers decades’ worth of work, but I was struck by a draw­ing of his from early in Xenakis’ ca­reer, when he was still work­ing in Le Corbusier’s stu­dio in Paris.

Xenakis facade drawings Iannis Xenakis, table with pro­gres­sions of rec­tan­gles with in­creas­ing widths drawn from the Modulor. Source: Fondation Le Corbusier, Paris. Reproduced in Sterken (2007).

It shows a se­ries of de­signs for the fa­cade of the monastery of Sainte-Marie de La Tourette in south­ern France. Each of the 18 de­signs is unique, yet they all seem to come from some com­mon sys­tem.

Xenakis facade drawings Iannis Xenakis and Le Corbusier: Sainte Marie de La Tourette. Source

My un­der­stand­ing is that he de­rives these from a har­moic se­ries based on the mod­u­lor se­ries. High-resolution scans of Xenakis’ draw­ings don’t seem to ex­ist (Even his own mono­graph suf­fers from poor re­pro­duc­tions), so it’s hard to re­verse-en­gi­neer the ex­act method he used to gen­er­ate these pat­terns. the best I can tell, he’s tak­ing one num­ber $x_0$, mul­ti­ply­ing it with an­other one (presumably the golden ra­tio $\varphi \approx 1.618034$, since that’s how the mod­u­lor is de­rived) to get a sec­ond one. The sec­ond num­ber is mul­ti­plied with $\varphi$ again to ob­tain a third, and the process is re­peated to cre­ate a se­ries of $n$ num­bers. In short:

$$x_n = x_{0} \varphi^n$$

Once the value of $x_n$ ex­ceeds a cer­tain thresh­old, he starts di­vid­ing by $\varphi$ in­stead of mul­ti­ply­ing. Once $x_n$ be­comes smaller than a cer­tain value, he switches back to mul­ti­pli­ca­tion, and so on. I think he re­peats this process for dif­fer­ent val­ues of $x_0$ and then com­bines the re­sult­ing se­ries of num­bers (by or­derubg them by value) to ob­tain the fi­nal re­sult.

The strik­ing thing about Xenakis’ method is that for all it’s math­e­mat­i­cal ex­act­ness and com­plex­ity, the re­sult feels en­tirely nat­ural. It is also worth point­ing out that he’s not re­ally de­sign­ing a fa­cade: His pro­ject is re­ally a ma­chine for mak­ing fa­cades. Between all the pos­si­ble val­ues of $x_0$, $n$ and $\varphi$, he opens up a fa­cade space filled with an in­fi­nite num­ber of points.

Xenakis’ al­go­rithm is rel­a­tively sim­ple — yet it al­lows for es­sen­tially in­fi­nite vari­a­tions. I saw in this a method to ex­plore the ten­sion be­tween line and tone in my un­der­grad ma­chine draw­ings in a struc­tured way.

I be­gan by writ­ing a tool that would al­low me to gen­er­ate vec­tor draw­ings fol­low­ing Xenakis’ al­go­rithm. I be­gin with a zig-zag line across the top of the page. For the sec­ond line, I take the num­ber of zig-zags in the first and mul­ti­ply it by a fixed ra­tio $\varphi$. The third line is de­rived by mul­ti­ply­ing the num­ber of zig-zags in the sec­ond with $\varphi$, and so on. Like Xenakis, I switch to di­vi­sion once a cer­tain thresh­old is reached.

Xenakis facade drawings

Xenakis (as far as I can tell) only used the Golden Ratio for his fa­cades. I like us­ing the 12 in­ter­vals in the har­monic scale, too.

Interval in C Ratio Ratio (1:x) % of larger value % of smaller value
uni­son C→C 1:1 1 100 100
mi­nor sec­ond C→D♭ 15:16 1.067 0.9372071228 106.7
ma­jor sec­ond C→D 8:9 1.125 0.8888888889 112.5
mi­nor third C→E♭ 5:6 1.2 0.8333333333 120
ma­jor third C→E 4:5 1.25 0.8 125
per­fect fourth C→F 3:4 1.333 0.7501875469 133.3
aug. fourth or dim. fifth C→F♯/G♭ 1:√2 1.414 0.7072135785 141.4
per­fect fifth C→G 2:3 1.5 0.6666666667 150
mi­nor sixth C→A♭ 5:8 1.6 0.625 160
ma­jor sixth C→A 3:5 1.667 0.599880024 166.7
mi­nor sev­enth C→B♭ 9:16 1.778 0.5624296963 177.8
ma­jor sev­enth C→B 8:15 1.875 0.5333333333 187.5
oc­tave C→C 1:2 2 0.5 200

Notes

  1. Owen Gregory (2011): Composing the New Canon: Music, Harmony, Proportion
  2. Sven Sterken (2007): Music as an Art of Space: Interactions be­tween Music and Architecture in the Work of Iannis Xenakis. Available at core.ac.uk/​down­load/​pdf/​34525212.pdf
  3. Alex Ross (2010): Waveforms: The sin­gu­lar Iannis Xenakis. The New Yorker. Available at newyorker.com/​mag­a­zine/​2010/​03/​01/​wave­forms