1、Fractal Music, ercards and More . . . Mathematical Recreations from SCIENTIFIC AMERICAN Magazine Fractal Music, Mathematical Recreations from SCIENTIFIC AMERICAN Magazine Martin Gardner W. H. Freeman and Company New York Library of Congress Cataloging-in-Publication Data Gardner, Martin, 1914- Fract
2、al music, hypercards and more : mathematical recreations from Scientific: American I by Martin Gardner. p. cn1. Includes index. ISBN 0-7 167-2 188-0. -ISBN 0-7 167-2 189-9(pbk.) 1. Mat hematical recreations. I. Scientific American. 11. Title. QA95.Gi16 1991 793.8-dc20 91-17066 CIP Copyright O 1992 b
3、y W. H. Freeman and Company No part of this book may be reproduced by any mechanical, photographic or electronic process, or in the form of a phonographic recording, nor may it be stored in a retrieval system, transmitted or otherwise copied for public or private use, without written permission from
4、 the publisher. Printed in the United States of America To Douglas Hofstadter For opening our eyes to the “strange loops“ inside our heads, to the deep mysteries of memory, intelligence, and self-awareness, and for the incomparable insights, humor, and wordplay in his writ- ings. Preface ix Chapter
5、One Chapter Two Chapter Three Chapter Four Chapter Five Chapter Six Chapter Seven Chapter Eight Chapter Nine Chapter Ten Chapter Eleven Chapter Twelve Chapter Thirteen Chapter Fourteen Chapter Fifteen Chapter Sixteen Chapter Senenteen Chapter Eighteen Chapter Nineteen Chapter Twenty Chapter Twenty-o
6、ne White, Brown, and Fractal Music 1 The Tinkly Temple Bells 24 Mathematical Zoo 39 Charles Sanders Peirce 61 Twisted Prismatic Rings 76 The Thirty Color Cubes 88 Egyptian Fractions 100 Minimal Sculpture 1 10 Minimal Sculpture I1 133 Tangent Circles 149 The Rotating Table and Other Problems 167 Does
7、 Time Ever Stop? Can the Past B;e Altered? 191 Generalized Ticktacktoe 202 Psychic Wonders and Probability 2 I14 Mathematical Chess Problems 228 Douglas Hofstadters adel, Escher, Bach 243 Imaginary Numbers 257 Pi and Poetry: Some Accidental Patterns 271 More on Poetry 281 Packing Squares 289 Chaitin
8、s Omega 307 Name Index 3 2 1 T his book reprints my Mathematical Games columns from the 1978 and 1979 issues of Scientific American Magazine. It is the fourteenth such collection, and I have one more to go be- fore running out of columns. As in previous anthologies, ad- denda to the chapters update
9、the material with information supplied by faithful readers, and by papers published after the columns were written. In two cases-a column c)n pi and poetry, and one on minimal sculpture-there was so much to add that I have written new chapters which appear here for the first time. The book is dedica
10、ted to my friend Douglas Hofstadter, whom I first met when he was seeking a publisher for his classic Godel, Escher, Bach, and who later became my suc- cessor at Scientific American. I had the privilege of reviewing GEB in a column that is reprinted here. kte, Brown, an d 1 Music Fracta “For when th
11、ere are no words accompanying music it is very difficult to recognize the meaning of the harmony and rhythm, or to see that any wor- thy object is imitated by them.“ -PLATO, Laws, Book I1 P lato and Aristotle agreed that in some fashion all the fine arts, including music, “imitate“ nature, and from
12、their day until the - late 18th century “imitation“ was a central concept in west- ern aesthetics. It is obvious how representational painting and sculpture “represent,“ and how fiction and the stage copy life, but in what sense does music imitate? By the mid-18th century philosophers and critics we
13、re still arguing over exactly how the arts imitate and .whether the term is relevant to music. The rhythms of music may be said to imitate such natural rhythms as heartbeats, walking, running, flapping wings, waving fins, water waves, the peri- odic motions of heavenly bodies and so on, but this doe
14、s not explain why we enjoy music more than, say, the souind of ci- cadas or the ticking of clocks. Musical pleasure derives mainly from tone patterns, and nature, though noisy, is singularly de- void of tones. Occasionally wind blows over some object to produce a tone, cats howl, birds warble, bowst
15、rings twang. A Greek legend tells how Hermes invented the lyre: he found a turtle shell with tendons attached to it that produced musical tones wlhen they were plucked. Above all, human beings sing. Musical instruments may be said to imitate song, but what does singing imitate? A sad, happy, angry o
16、r serene song somehow resembles sadness, joy, anger or serenity, but if a melody has no words and invokes no special mood, what does it copy? It is easy to understand Platos mystification. There is one exception: the kind of imitation that plays a role in “program music.“ A lyre is severely limited
17、in the nat- ural sounds it can copy, but such limitations do not apply to symphoic or electronic music. Program music has no diffi- culty featuring the sounds of thunder, wind, rain, fire, ocean waves and brook murmurings; bird calls (cuckoos and crow- ing coclrs have been particularly popular), fro
18、g croaks, the gaits of animals (the thundering hoofbeats in Wagners Ride of the Valkyries), the flights of bumblebees; the rolling of trains, the clang of hammers; the battle sounds of marching soldiers, clashing: armies, roaring cannons and exploding bombs. S1aughtc.r on Tenth Avenue includes a pis
19、tol shot and the wail of a police-car siren. In Bachs Saint Matthew Passion we hear the earthquake and the ripping of the temple veil. In the Al- pine Syvnphony by Richard Strauss, cowbells are imitated by the shaking of cowbells. Strauss insisted he could tell that a certain female character in Fel
20、ix Mottls Don Juan had red hair, and he once said that someday music would be able to distinguish the clattering of spoons from that of forks. Such imitative noises are surely a trivial aspect of music even when it accompanies opera, ballet or the cinema; be- sides, siuch sounds play no role whatsoe
21、ver in “absolute mu- sic,“ music not intended to “mean“ anything. A Platonist might argue that abstract music imitates emotions, or beauty, or the divine harmony of God or the gods, but on more mundane lev- els musilc is the least imitative of the arts. Even nonobjective paintings resemble certain p
22、atterns of nature, but nonobjec- tive music resembles nothing except itself. Since the turn of the century most critics have agreed that “imitation“ has been given so many meanings (almost all are found in Plato) that it has become a useless synonym for “resemblance.“ When it is made precise with re
23、ferenlce to lit- erature or the visual arts, its meaning is obvious and trivial. When it is applied to music, its meaning is too fuzzy to be helpful. In this chapter we take a look at a surprising discov- ery by Richard F. Voss, a physicist from Minnesota who joined the Thomas J. Watson Research Cen
24、ter of the International Business Machines Corporation after obtaining his 1Ph.D. at the University of California at Berkeley under the guidance of John Clarke. This work is not likely to restore “innitation“ to the lexicon of musical criticism, but it does suggest a cu- rious way in which good musi
25、c may mirror a subtle statistical property of the world. The key concepts behind Vosss discovery are what math- ematicians and physicists call the spectral density (or power spectrum) of a fluctuating quantity, and its “autocorrelation.“ These deep notions are technical and hard to understand. Be- n
26、oit Mandelbrot, who is also at the Watson Research Center, and whose work makes extensive use of spectral densities and autocorrelation functions, has suggested a way of avoiding them here. Let the tape of a sound be played faster or slower than normal. One expects the character of the sound to chan
27、ge considerably. A violin, for example, no longer sounds like a violin. There is a special class of sounds, however, that be- have quite differently. If you play a recording of such a sound at a different speed, you have only to adjust the volume to make it sound exactly as before. Mandelbrot calls
28、sucl sounds “scaling noises.“ By far the simplest example of a scaling noise is what in electronics and information theory is called white noise (or “Johnson noise“). To be white is to be colorless. White noise is a colorless hiss that is just as dull whether you play it faster or slower. Its autoco
29、rrelation function, which measures how its fluctuations at any moment are related to previous fluctua- tions, is zero except at the origin, where of course it must be 1. The most commonly encountered white noise is the ther- mal noise produced by the random motions of electrons through an electrical
30、 resistance. It causes most of the static in a radio or amplifier and the “snow“ on radar and television screens when there is no input. With randomizers such as dice or spinners it is easy to generate white noise that can then be used for composing a random “white tune,“ one with no correlation bet
31、ween any White, Brown, and Fractal Music 3 two notes. Our scale will be one octave of seven white keys on a piano: do, re, me, fa, so, la, ti. Fa is our middle fre- quency. Now construct a spinner such as the one shown at the left in Figure 1. Divide the circle into seven sectors and label them with
32、 the notes. It matters not at all what arc lengths are assigned to these sectors; they can be completely arbi- trary. On the spinner shown, some order has been imposed by giving fa the longest arc (the highest probability of being chosen) and assigning decreasing probabilities to pairs of notes that
33、 are equal distances above and below fa. This has the ef- fect of clustering the tones around fa. To produce a “white melody“ simply spin the spinner as often as you like, recording each chosen note. Since no tone is related in any way to the sequence of notes that precedes it, the result is a total
34、ly uncorrelated sequence. If you like, you can divide the circle into more parts and let the spinner select notes that range over the entire piano keyboard, black keys as well as white. To nake your white melody more sophisticated, use an- other spinner, its circle divided into four parts (with any
35、pro- portions you like) and labeled 1, 112, 114 and 118 so that you can assign a full, a half, a quarter or an eighth of a beat to each tone. After the composition is completed, tap it out on the piano. The music will sound just like what it is: random music of the dull kind that a two-year-old or a
36、 monkey might produce by hitting keys with one finger. Similar white music can be lbased on random number tables, or the digits in an irrational number. A rrlore complicated kind of scaling noise is one that is sometimes called Brownian noise because it is characteristic of Browinian motion, the ran
37、dom movements of small particles suspended in a liquid and buffeted by the thermal agitation of molecules. Each particle executes a three-dimensional “ran- dom walk,“ the positions in which form a highly correlated sequencle. The particle, so to speak, always “remembers“ where it has been. When tone
38、s fluctuate in this fashion, let us follow Voss and call it Brownian music or brown music. We can produce it easily with a spinner and a circle divided into seven parts as before, but now we label the regions, as shown at the right in Figure 1, to represent intervals between successive tones. These
39、step sizes and their probabilities can be whatever we like. On the spinner shown, plus means a step up the scale of FIGURE 1 Spinners for white music (left) and brown music (right) white, Brown, and Fractal Music one, two or three notes and minus means a step dourn of the same intervals. Start the m
40、elody on the pianos middle C, then use the spinner to generate a linear random walk up and down the keyboard. The tune will wander here and there, ,and will eventually wander off the keyboard. If we treat the ends of the keyboard as “absorbing barriers,“ the tune ends when we encounter one of them.
41、We need not go into the ways in which we can treat the barriers as reflecting barriers, allowing the tune to bounce back, or as elastic barriers. To make the bar- riers elastic we must add rules so that the farther the tone gets from middle C, the greater is the likelihood it will step back toward C
42、, like a marble wobbling from side to side as it rolls down a curved trough. As before, we can make our brown music more sophisti- cated by varying the tone durations. If we like, we can do this in a brown way by using another spinner to give not the dura- tion but the increase or decrease of the du
43、ration-another random walk but one along a different street. The result is a tune that sounds quite different from a white tune because it is strongly correlated, but a tune that still has little alesthetic appeal. It simply wanders up and down like a drunk weaving through an alley, never producing
44、anything that resembles good music. If we want to mediate between the extremes of white and brown, we can do it in two essentially different ways. The way chosen by previous composers of “stochastic music“ is to adopt transition rules. These are rules that select each note 5 on the basis of the last
45、 three or four. For example, one can analyze Bachs music and determine how often a certain note follows, say, a certain triplet of preceding notes. The random selectiori of each note is then weighted with probabilities de- rived from a statistical analysis of all Bach quadruplets. If there are certa
46、in transitions that never appear in Bachs mu- sic, we add rejection rules to prevent the undesirable transi- tions. The result is stochastic music that resembles Bach but only superficially. It sounds Bachlike in the short run but ran- dom in the long run. Consider the melody over periods of four or
47、 five notes and the tones are strongly correlated. Compare a run of five notes with another five-note run later on and you are back to white noise. One run has no correlation with the other. Almost all stochastic music produced so far has been of this sort. It sounds musical if you listen to any sma
48、ll part but random and uninteresting when you try to grasp the pat- tern as a whole. Vosss insight was to compromise between white and brown input by selecting a scaling noise exactly halfway be- tween. 1.n spectral terminology it is called llf noise. (White noise has a spectral density of llf O, br
49、ownian noise a spectral density 13f llf 2. In “one-over-f“ noise the exponent off is 1 or very close to 1.) Tunes based on llf noise are moderately cor- related, not just over short runs but throughout runs of any size. It turns out that almost every listener agrees that such music is much more pleasing than white or brown music. In electronics llf noise is well known but poorly under- stood. It is sometimes called flicker noise. Mandelbrot, whose book Th!e Fractal Geometry of Nature (W. H. Freeman and Company, 1982) has already become a modern classic, was the firs
