From decwrl!uunet!allbery Thu Apr 19 05:38:54 PDT 1990 Article 1491 of comp.sources.misc: Path: decwrl!uunet!allbery From: edgar@shape.mps.ohio-state.edu (Gerald Edgar) Newsgroups: comp.sources.misc Subject: v12i011: Dragon curves with music Message-ID: <85035@uunet.UU.NET> Date: 18 Apr 90 00:45:24 GMT Sender: allbery@uunet.UU.NET Organization: The Ohio State University, Dept. of Math. Lines: 415 Approved: allbery@uunet.UU.NET (Brandon S. Allbery - comp.sources.misc) Posting-number: Volume 12, Issue 11 Submitted-by: edgar@shape.mps.ohio-state.edu (Gerald Edgar) Archive-name: music.logo/part01 Dragon curves with music. Explanation and Logo source code. #! /bin/sh # This is a shell archive. Remove anything before this line, then unpack # it by saving it into a file and typing "sh file". To overwrite existing # files, type "sh file -c". You can also feed this as standard input via # unshar, or by typing "sh 'music.doc' <<'END_OF_FILE' X X ***** DRAGON CURVES WITH MUSIC ***** X XMany of the "fractal" or "self-similar" curves (I call them dragon curves) Xcan be generated by recursive turtle graphics routines. The basic Xinstructions in turtle graphics are: X FORWARD :D go forward for a distance :D X RIGHT :X turn right by :X degrees X LEFT :Y turn left by :Y degrees XFor example, here is the program for the Koch curve: X X TO KOCH :SIZE :DEPTH X IF :DEPTH=0 [FORWARD :SIZE STOP] X KOCH :SIZE/3 :DEPTH-1 X LEFT 60 X KOCH :SIZE/3 :DEPTH-1 X RIGHT 120 X KOCH :SIZE/3 :DEPTH-1 X LEFT 60 X KOCH :SIZE/3 :DEPTH-1 X END X XIn this program, and the other recursive programs that construct dragon Xcurves, there is a parameter :DEPTH. When :DEPTH is zero, the curve Xis just a line segment. When :DEPTH is increased by one, each segment Xis replaced by a copy of a basic figure consisting of a few line segments. XThe lengths of the individual segments are decreased when this happens, Xof course. X XThe drawing commands are simply moving forward by a certain distance, Xand turning by a certain angle. This suggests a musical accompaniment Xfor the drawing of such dragon curves. Moving forward for a distance :D Xwill correspond to playing a note for length of time :D. Turning will Xcorrespond to changing the pitch of the note. (Right = higher, Xleft = lower.) The natural correspondence between angles and changes in Xpitch lets an octave change correspond to 360 degrees turn. Thus, a turn Xby :D degrees means that the frequency is multiplied by the factor X2^(:D/360). For example, a 60-degree turn is a full step; a right-angle Xturn is a minor third; a 120-degree turn is a major third; and so on. XNote that the dragon "McWorter's Pentigree" uses angles of 72 degrees; Xthis interval, 2^(1/5), is not used in Western music. Do you like the Xsound? X XIn the programs, the length of the note played is simply proportional to Xthe length of the line segment. But a more mathematically correct system Xwould be to make the time to traverse the whole curve remain constant Xwhen :DEPTH is changed. (So the duration of the note is proportional Xto a power of the length of the segment. The exponent used here is Xthe "fractal dimension".) For example, it the Koch curve, above, Xthe fractal dimension is s = (log 4)/(log 3). So when :DEPTH is Xincreased by 1, the length of the line segments is decreased by Xfactor 1/3, and the length of the notes should be decreased by factor X(1/3)^s = 1/4. The number of line segments is increased by a factor of X4, so the total playing time is unchanged. X XThe actual "fractal" is the self-similar curve that is the limit as X:DEPTH increases without bound. In practice, when :DEPTH gets large Xenough, the pictures no longer change; so the picture gives us a good Xidea of the fractal curve itself. An interesting experiment would be Xto do this with the music. I have not been able to reach this goal on my XMacintosh Plus, since I am using a fairly slow Logo, and since the Mac Xwill not play notes shorter than a certain length. X X(There is some literature on "fractal music". It is NOT this.) X X-------------------------------------------------------------------- X Gerald A. Edgar April, 1990 X Dept. of Mathematics X The Ohio State University X Columbus, OH 43210 70715,1324 (CompuServe) X edgar@mps.ohio-state.edu (internet) EDGAR@OHSTPY (bitnet) END_OF_FILE if test 3435 -ne `wc -c <'music.doc'`; then echo shar: \"'music.doc'\" unpacked with wrong size! fi # end of 'music.doc' fi if test -f 'music.logo' -a "${1}" != "-c" ; then echo shar: Will not clobber existing file \"'music.logo'\" else echo shar: Extracting \"'music.logo'\" \(6588 characters\) sed "s/^X//" >'music.logo' <<'END_OF_FILE' X X;; Dragon curves with music. Logo source code. X X;; The curves are: X;; 1 McWorter's Pentigree X;; 2 2 Dimensional Curve X;; 3 Triadic (SNOWFLAKE) Curve X;; 4 Quadric Curve X;; 5 Pinwheel Curve X;; 6 Hexomino Curve X;; 7 Gosper Curve (FLOWSNAKE) X;; 8 Arrow Curve X;; 9 Peano Curve X X;; Sample usage: FRACTAL6 100 2 X;; This draws the Hexomino curve with :size 100 and :depth 2 X;; :size is the size of the curve, X;; :depth is the complexity of the curve (simplest is 0) X X;; Or try my preset values to start with: for example X;; FRACTALS 6 for the Hexomino Curve. (Replace 6 by any X;; digit 1 to 9 for the other curves.) X X;; --------------------------------------------------------------- X X;; Tested on a Macintosh Plus using Coral Object Logo. X;; For other versions of Logo, you may need to make changes. X;; For example the following: X X;; CS X;; clears the screen. It may be called CG; if so, define it: X;; TO CS X;; CG X;; END X X;; TOOT :F :A :D X;; produces a note with frequency :F Hertz; X;; amplitude :A (0 to 255); duration :D seconds X;; Toot is called only by TOOTX with a duration :X/:DENOM, X;; so the speed can be adjusted by changing the value of X;; "DENOM. If your TOOT has duration measured in 60ths of X;; a second, then redefine TOOTX as follows: X;; TO TOOTX :X X;; TOOT :FREQ 100 :X/(:DENOM/60) X;; END X X;; The starting frequency is :FREQ, so the pitch can be adjusted X;; by changing the value of "FREQ. X X;; G. A. Edgar April, 1990 X;; Dept. of Mathematics X;; The Ohio State University X;; Columbus, OH 43210 X;; edgar@mps.ohio-state.edu (internet) EDGAR@OHSTPY (bitnet) X X;; --------------------------------------------------------------- X Xmake "genlist [3 5 4 4 3 3 3 3 3] Xmake "denomlist [200 175 50 50 100 80 150 80 65] Xmake "denom 50 Xht cs X Xto fractal1 :size :depth Xif :depth = 0 [forward :size tootx :size stop] Xleftmusic 36 Xfractal1 :size/2.62 :depth-1 Xleftmusic 72 Xfractal1 :size/2.62 :depth-1 Xrightmusic 144 Xfractal1 :size/2.62 :depth-1 Xrightmusic 72 Xfractal1 :size/2.62 :depth-1 Xleftmusic 72 Xfractal1 :size/2.62 :depth-1 Xleftmusic 72 Xfractal1 :size/2.62 :depth-1 Xrightmusic 36 Xend X XTO FRACTAL2 :SIZE :DEPTH XIF :DEPTH = 0 [FD :SIZE TOOTX :SIZE STOP] XFRACTAL2 :SIZE / 2 :DEPTH - 1 Xleftmusic 90 XFRACTAL2 :SIZE / 2 :DEPTH - 1 Xrightmusic 180 XFRACTAL2 :SIZE / 2 :DEPTH - 1 Xleftmusic 90 XFRACTAL2 :SIZE / 2 :DEPTH - 1 XEND X XTO FRACTAL3 :SIZE :DEPTH XIF :DEPTH = 0 [FD :SIZE TOOTX :SIZE STOP] XFRACTAL3 :SIZE / 3 :DEPTH - 1 Xleftmusic 60 XFRACTAL3 :SIZE / 3 :DEPTH - 1 Xrightmusic 120 XFRACTAL3 :SIZE / 3 :DEPTH - 1 Xleftmusic 60 XFRACTAL3 :SIZE / 3 :DEPTH - 1 XEND X XTO FRACTAL4 :SIZE :DEPTH XIF :DEPTH = 0 [FD :SIZE TOOTX :SIZE STOP] XFRACTAL4 :SIZE / 3 :DEPTH - 1 Xleftmusic 90 XFRACTAL4 :SIZE / 3 :DEPTH - 1 Xrightmusic 90 XFRACTAL4 :SIZE / 3 :DEPTH - 1 Xrightmusic 90 XFRACTAL4 :SIZE / 3 :DEPTH - 1 Xleftmusic 90 XFRACTAL4 :SIZE / 3 :DEPTH - 1 XEND X XTO FRACTAL5 :SIZE :LEVEL XPINWHEEL :SIZE :LEVEL XEND X XTO FRACTAL6 :SIZE :LEVEL XHEX :SIZE :LEVEL XEND X XTO FRACTAL7 :SIZE :DEPTH XFLOW 0.9 * :SIZE "TRUE :DEPTH XEND X XTO FRACTAL8 :SIZE :LEVEL XARROW :SIZE :LEVEL XEND X XTO FRACTAL9 :SIZE :LEVEL XPEANO 0.9 * :SIZE :LEVEL 1 XEND X XTO FRACTALS :TYPE XSETUP Xmake "denom item :type :denomlist XRUN ( SE (WORD "FRACTAL :type) 200 (item :type :genlist) ) XEND X XTO TOOTX :X Xtoot :freq 100 :X/:denom XEND X XTO PEANO :S :LEVEL :P XLOCAL "A XIF :LEVEL = 0 [FD :S TOOTX :S STOP] XMAKE "A 0.25 * 2.236067977 * :S Xleftmusic 60 * :P XPEANO :S / 3 :LEVEL - 1 -:P XPEANO :S / 3 :LEVEL - 1 :P Xrightmusic 60 * :P XPEANO :S / 3 :LEVEL - 1 :P Xrightmusic 60 * :P XPEANO :S / 3 :LEVEL - 1 :P Xrightmusic 150 * :P XPEANO :A / 3 :LEVEL - 1 :P XPEANO :A / 3 :LEVEL - 1 -:P Xleftmusic 60 * :P XPEANO :A / 3 :LEVEL - 1 -:P Xleftmusic 60 * :P XPEANO :A / 3 :LEVEL - 1 -:P Xleftmusic 90 * :P XPEANO :S / 3 :LEVEL - 1 :P Xrightmusic 150 * :P XPEANO :A / 3 :LEVEL - 1 :P XPEANO :A / 3 :LEVEL - 1 -:P Xleftmusic 150 * :P XPEANO :S / 3 :LEVEL - 1 -:P XPEANO :S / 3 :LEVEL - 1 :P XEND X XTO SETUP XCS XPU SETPOS [-100 -20] PD XSETH 90 Xmake "freq 440 XEND X XTO 7LB :N :DIR :L Xrightmusic 79.107 XFLOW :N NOT :DIR :L Xleftmusic 60 FLOW :N :DIR :L XFLOW :N :DIR :L Xleftmusic 120 FLOW :N :DIR :L Xleftmusic 60 FLOW :N NOT :DIR :L Xrightmusic 120 FLOW :N NOT :DIR :L Xrightmusic 60 FLOW :N :DIR :L Xleftmusic 19.107 XEND X XTO 7LF :N :DIR :L Xrightmusic 19.107 XFLOW :N :DIR :L Xleftmusic 60 FLOW :N NOT :DIR :L Xleftmusic 120 FLOW :N NOT :DIR :L Xrightmusic 60 FLOW :N :DIR :L Xrightmusic 120 FLOW :N :DIR :L XFLOW :N :DIR :L Xrightmusic 60 FLOW :N NOT :DIR :L Xleftmusic 79.107 XEND X XTO FLOW :N :DIR :L XIF :L = 0 [FD :N TOOTX :N STOP] XIFelse :DIR [7LF :N / SQrt 7 :DIR :L - 1] [7LB :N / SQrt 7 :DIR :L - 1] XEND X XTO FLOWSNAKE Xcs ht XPU SETPOS [30 -20] PD XREPEAT 3 [FLOW 90 "TRUE 3 rightmusic 120] Xleftmusic 120 XFLOW 90 "TRUE 3 Xrightmusic 120 XFLOW 90 "TRUE 3 XFLOW 90 "TRUE 3 XEND X XTO HEX :SIZE :LEVEL XIF :LEVEL = 0 [FD :SIZE TOOTX :SIZE STOP] XHEX 0.25 * :SIZE :LEVEL - 1 Xleftmusic 60 XHEX 0.25 * :SIZE :LEVEL - 1 Xrightmusic 60 XHEX 0.25 * :SIZE :LEVEL - 1 Xrightmusic 60 XHEX 0.25 * :SIZE :LEVEL - 1 Xrightmusic 120 XHEX 0.25 * :SIZE :LEVEL - 1 XHEX 0.25 * :SIZE :LEVEL - 1 Xleftmusic 120 XHEX 0.25 * :SIZE :LEVEL - 1 Xleftmusic 60 XHEX 0.25 * :SIZE :LEVEL - 1 Xleftmusic 60 XHEX 0.25 * :SIZE :LEVEL - 1 Xrightmusic 60 XHEX 0.25 * :SIZE :LEVEL - 1 XEND X XTO PINWHEEL :SIZE :LEVEL XIF :LEVEL = 0 [FD :SIZE TOOTX :SIZE STOP] Xleftmusic 30 XPINWHEEL 0.288675 * :SIZE :LEVEL - 1 Xrightmusic 60 XPINWHEEL 0.288675 * :SIZE :LEVEL - 1 Xleftmusic 60 XPINWHEEL 0.288675 * :SIZE :LEVEL - 1 Xleftmusic 120 XPINWHEEL 0.288675 * :SIZE :LEVEL - 1 Xleftmusic 120 XPINWHEEL 0.288675 * :SIZE :LEVEL - 1 Xleftmusic 0 XPINWHEEL 0.288675 * :SIZE :LEVEL - 1 Xrightmusic 120 XPINWHEEL 0.288675 * :SIZE :LEVEL - 1 Xrightmusic 120 XPINWHEEL 0.288675 * :SIZE :LEVEL - 1 Xrightmusic 60 XPINWHEEL 0.288675 * :SIZE :LEVEL - 1 Xleftmusic 60 XPINWHEEL 0.288675 * :SIZE :LEVEL - 1 Xrightmusic 30 XEND X XTO ARROW :SIZE :LEVEL XIF :LEVEL = 0 [FD :SIZE TOOTX :SIZE STOP] XARROW 0.25 * :SIZE :LEVEL - 1 XARROW 0.25 * :SIZE :LEVEL - 1 Xleftmusic 120 XARROW 0.25 * :SIZE :LEVEL - 1 Xrightmusic 120 XARROW 0.25 * :SIZE :LEVEL - 1 Xrightmusic 60 XARROW 0.25 * :SIZE :LEVEL - 1 Xrightmusic 120 XARROW 0.25 * :SIZE :LEVEL - 1 Xleftmusic 60 XARROW 0.25 * :SIZE :LEVEL - 1 Xleftmusic 120 XARROW 0.25 * :SIZE :LEVEL - 1 Xleftmusic 60 XARROW 0.25 * :SIZE :LEVEL - 1 Xrightmusic 60 XARROW 0.25 * :SIZE :LEVEL - 1 XEND X Xto leftmusic :x Xleft :x Xmake "freq :freq/power 2 (:x/360) Xend X Xto rightmusic :x Xright :x Xmake "freq :freq*power 2 (:x/360) Xend X X END_OF_FILE if test 6588 -ne `wc -c <'music.logo'`; then echo shar: \"'music.logo'\" unpacked with wrong size! fi # end of 'music.logo' fi echo shar: End of shell archive. exit 0 -- Gerald A. Edgar Department of Mathematics Bitnet: EDGAR@OHSTPY The Ohio State University Internet: edgar@mps.ohio-state.edu Columbus, OH 43210 ...!{att,pyramid}!osu-cis!shape.mps.ohio-state.edu!edgar