Drawings based on laser lissajous figures and the lambdoma diagram

and the lambdoma diagram Leonardo mund In: - 11 - 1978 - p. 301-303 SCS & AERS ®- AS one 1818 1 DRAUTURS ERATED OM LALE HEF DO. Ferrara Leoreenae Hls mutiny Lamis Document ITS) “eh TRA Zit— = ares CDLori DIAGRAMS À TONE st In Ems tem Trees Arm TIG Pre rents FB V'IELAL PELATIQUEMIF THE MAPMOTT N SERIES, THe COMFODEITIOHE mature THE TO THE STRUCTURE ETART ING FOINT SENERATION OF MUSICAL OF MUIICAL FOR EOTH TONES THE Wet TH MATPI CES OF THROWER PICTORIAL u THA ORE rs Diez or NORM AS THE L Mi. Tre INTERV ALT HEPE SAL IPRPATED Grit SENEPATEL Fv MEANT OF ELECTRONIC ECUIFMEN T. WIEN A LASER Ant À SCANNING DEVICE WERE ADDED To THE ENETEM Lirsksour FIGURES FORMED EFS THE QFCILLATIONS OF THE TONES HERE FROJECTE L. THAT THE CHANGES IM IHAFEZ OF THE FIGURES WHERE DIE TO THE FEATS OF THE TONES: INDICATE THAT FEATS HERE ACOUSTICAL PATHER THAN A FEVCHOLDFICHL PHENOMENO N: ZURSESTINS THAT THE‘ FOUT PE IMWEETIGATEL TUTERVGELS WERE PRESENTED or THE COMPONENTS OF THESE FIGURES SERVED AZ LinsPAne AE ÄMERICAN-FAZEL Ott BTEFN APT -FICTOPIAL JEJECT NATIONALITY? SUEJELT STYLE: THE TO ADD TO THE WELL AS THE LAFER ZONE rr re ee UALERETANDINES LISSRJOQUE INFFIPATION FOR OTHER LRAUIMES, ur TlezcriPToORS:? DRAWING» LAMEDOMA DIAGRAM FURTHER, Ih À LISUAL ARRAY THE FIeupes. AUTHOR: FILA LissaJous FIGURES Art ABTS, AMERICAN MELIUMme Form? prewreas Haris Dreurtors RJOLF FIGUPES GHD THE} LAMELONR Liespen HERD: Ear Pyrhdgorcan] LeonePng ILLUETPATIONS tente USE Ed Dowumeest FREFENTE THE | COLOR! DPresRars 13H T A tE HepnDrit “POMP DFITION: L PELATIQNEMIF FERIEF. THE AND TO THE ETRUUITUPE FTARTIUF FONT OF À FOR MOST RL TOME TRO FOT TRE FITTORIML TRE GENERATION OF MUSICAL TONES War THe Fo rHasorEess. ENDE ss THE Or. THE INTERVALE WHERE -MLIEPMTED : FENERSTEL Ey meer OF ELECTRONIC EPUIFMEMTS WEN mM LASER AUD À ERA rn DEVICE WE RPE RODE TL TO THE FvETem L lESARJOUE FIGURES PFEILLATIONE OF THE TONES WERE FROSJECTED., THat THE THE FIGURES WEPE LUE TO THE BEATS OF THE TONHEZ: FORMED Fı THE THENGEF IHBICIATE Itt ZHAFES OF Tar FEATS GERE IEA RETHER THAN À FECCHOLGSICHL FMENONENDMN: Ft EFT Ius THAT THEY FMOQULE FE INVEETIGATED FURTHER. THE HATPICEE OF CHE HIT Icey . PUTEPUELE WERE FRESENTEL In A rsa ARPA TO ADD TO THE LIIDERET Erin? AE VELL Ar TRE LISFeJOUS EF Tzr DF THE el eutR mere TOME DHENTE THE nr FILA: THEZE FIGURES SERVED Af INfFIPATIQN FOR DTHER DPAUINES. "RUTHOR« Lik DEECRIETDPES LPALIIGe ANERICARN- RASE Di: LAZER RJONE FISUPRESZS este LAHELDNME Th EPE LECTION HEADING! sz): ST-MODEPN AFT-ereroPiat APTE TUEJECT NATIONALITY? American TUEIJEST STYLE: Mentone Forest operas

Author(s): Barbara Hero Source: Leonardo, Vol. 11, No. 4 (Autumn, 1978), pp. 301-303 Published by: The MIT Press Stable URL: http://www.jstor.org/stable/1573956 Accessed: 19/01/2010 02:44 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=mitpress. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. The MIT Press is collaborating with JSTOR to digitize, preserve and extend access to Leonardo. http://www.jstor.org

Leonardo,Vol. 11, pp. 301-303. c PergamonPressLtd. 1978. Printedin GreatBritain. 1001-0301$02.00/0 DRAWINGS BASED ON LASER LISSAJOUS FIGURES AND THE LAMBDOMA DIAGRAM Barbara Hero* 1. recorded inputs. Two outputs from the amplifier, corresponding to X and Y, drive two galvanometers for laser scanning[3] (General Scanning, Watertown, Mass.). One, rotating on a vertical axis, carries a mirror 7 x 7 mm; the other, rotating on a horizontal axis, carries a mirror 7 x 11 mm. The mirrors oscillate to the beats of the different frequencies received. The helium-neon laser beam impinges on one mirror, is reflected to the next and is finally reflected to a wall or a projection screen. The power of a laser (0.5 mW) is sufficient to produce a satisfactory image on a wall at a distance of 3-4.5 m. I calibrate the system so that the Lissajous figures can be analyzed, although I do not attempt to do so myself. In an earlier article in Leonardo, I described some nonfigurative paintings that I made whose patterns and colors were related in arbitrary ways to relative pitch in music [1]. A more extensive discussion of my work relating visual art and music is given in my book, Eyes + Ears = Ideas[2]. The starting point for the pictorial compositions is the Lambdoma or Pythagorian diagram, which dates back to Ancient Greece. For recent works I use the following procedure: I select a number of tones with the aid of the Lambdoma diagram, produce the tones electronically and make sketches of their Lissajous figures, which are produced by means of a helium-neon laser scanner device and projected onto a wall or screen. Finally, I make a drawing based on the Lissajous figures with colors chosen according to an arbitrary code that identifies colors with 3. My aim is to produce Lissajous figures that are aesthetically pleasing as well as meaningful. It is desirable to minimize dissonances to obtain well delineated designs. The absence of dissonance is assured when only frequencies of the harmonic series are employed. The numerical values of the frequencies of the harmonic series are obtained by multiplying the frequency of the fundamental tone (which I have chosen to be 352 Hz) by the integers 1-16 and by dividing each of the products by the integers 1-16. This calculation, which is realistically restricted to integral multipliers and divisors below 17 to provide frequencies covering eight octaves and which roughly includes the frequency range of present-day pianos, follows the plan of the Lambdoma diagram [1]. The results of the calculation can be tabulated conveniently as an array of 256 numbers (Fig. 1). The notation for the harmonic series is shown below the array. Figure 1 shows the tone notations corresponding to each of the frequencies in the array. To determine the frequency of a tone, one must read its coordinates x and y; then multiply 352 by x and divide the result by y. For example for g, read x = 4 and y = 7. The frequency is then calculated to be 201 Hz. In Fig. 1, each of the 16 horizontal rows represents one overtone of the harmonic series and each of the 16 vertical columns represents one undertoneof the series. For convenience the first of the harmonic series begins with the fundamental f1l (44) and is followed by f, (88), an octave above it, then by the fifth c (132), the third octave f (176), the third a (220), another fifth c1 (264), the seventh e b (308), the octave fl (352), g1 (396), a1 (440), bl (484), c11 (528), d11 (572), e bl1 (616), e11 (660) and f11 (704). The undertone series can be traced in the same way downwards for four octaves, starting at f1 (352) or f11 (704) up to f" (5632). When a pure sine-wave tone is mixed with another, a sum and a difference are produced. For example, if fl (352 Hz) is combined with its third harmonic cT1 (1056 Hz), frequencies. 2. A tone of a specific frequency is produced by an electronic sine-wave oscillator (Eico 379 solid state sine/square wave generator, Eico Electronic Instrument Co, Brooklyn, N.Y.). Sinusoidal wave forms may be produced in a frequency range from 20 Hz to 2m Hz, but I do not exceed 1200 Hz. It is essential in my work that tones have wave forms with a minimum of overtones of the harmonic series. Therefore sinusoidal wave forms are preferred to square and triangular ones. I record the frequencies of the harmonic tones from the Lambdoma diagram (Fig. 1), one of which is a fundamental, separately on each of two channels of a magnetic tape. Later, using a second tape recorder, I make mixtures of these tones, which I record on one channel of a magnetic tape and different mixtures of tones on the second channel. These two channels are used as input to the scanners that produce the Lissajous figures [3,4]. The production of Lissajous figures in response to music by means of a helium-neon laser beam scanner for projection on a screen has been described in Leonardoby Wagler [5]. He reported on the use of a Sonavision projector manufactured by Sonavision, Ann Arbor, Michigan. I employ a Metrologic laser (model ML-600, 0.5 mw, Metrologic Instruments, Bellmaur, N.J.). The tape-recorded tones on two channels, called X and Y are used as an input to an amplifier (Studio Standard CA-2100, Fisher, Long Island City, N.J.) having four outputs, two of which are speakers, enabling one to hear as well as to see simultaneously the sum of the two tape*Artist, 48 LawrenceStrret, Boston, MA 02116, U.S.A. (Received 19 Nov. 1977.)

Barbara Hero (O V ER TONES) X 2 1 352 1 fI 3 88 f, 4 z 7 0 8 W 9 z 10 " 6 7 8 11 9 10 12 13 14 15 16 704 1056 1408.176a2112 2464 2816 318635203872 4224 4576 4928 5280 5632 fli c111 fill al' civ ekV fiv g"~ av bv c& dvX #b ev fv 176 264 352 440528 616 704 792880968 1056 1144123213201408 cl f' a, c"l ei, f1 g"l ai" b& c"' dW e1 e'" t'*' f 70 140 210 280 350 420 490 560 630l 700 770 840 910 980 1050 1120 C ci db b f" gU a~" al a! b b d f1eIb dl dOd' ad 928 696 870 754 464 812 638 522 496 580 116 290 232 174 348 58 b fil f,'r g' a" bc f bi? d< f1 9>' b1 c11 d<A^ b0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 d' t' g' a' b' ct'" d" e<c fl' fail g* Pi g, d g b 44 88 132 176 220 264 308 352 396 4340 484 528 572 616 660 704 b<l c"l d<" ei eI1 fit a co e~t fl g, a f c f f, 39 78 117 156 195 234 273 312 351 390 429 468 507 546 585 624 a c> dst fi g<c a}> a' b>' c>" d-< d> d> d> a~ d> g< 35 70 105 140 175 210 245 280 315 350 385 420 455 490 525 560 a d I eP f' g9< aK a>1 bcl clt dbii b f dc d , a, d 512 32 64 96 128 160 192 224 256 288 320 352t 384 416 448.480 a~> bk< b gl< a e< g< a> b> d'< e g< bji b? b6 29 58 87 116 145 174 203 232261 290 319 348 377 406 435 464 ba g> bh cI dtc e4t f fO g > at bJ d< f bti' fg pt 405 378 351 297 324 243 270 216 189 108 135 432 162 81 27 54 e2 f4 ft g> a: a< a< e< a<1 c e< fX a< b< cl di 325 350 375 400 275 300 225250 175 200 150 125 100 25 50 75 g9 b cot di f e<f f g a g94 b d gt gli d 368 345 276 299 322 253 23Q 138 161 184 207 92 i1 69 46 23 e d c f f aa fl bl c> e< f4 c> f> 22 44 66 88 110 132 154 176 198 220 242 264286 308330 352 e d e b< c c< ft a c e f 9 f f f a, fl 5 6 5 4 2816 176 352 528 704 880 105612321408 158417601936 2112 228824642640 cil f" a" c"' e ' fill gil"' a"' 13 civ dv< e J el flv fI f 117 234 351 468 585 702 819 936 10531170128714041521163817551872 bb dk f" 9g>" b'l c"il dk' e P' f111f19 g%' a{" bV" bt f' b~ 2 (m 3 11 >. 12 13 14 15 16 ,o4)o -);, f' g' a' b C a e 308 352 3964404845285t2 FFC1f76'722 Fc 4F1 264 e' et 616 660 f 704 HARM O NI C S E R I E S an assumedfundamenta f (352 Fig. 1. Musical tones within8 octaves (4 octaves below and 4 octaves above in the Lambdomadiagram. sound frequencies the to placements an of in corresponding array arranged Hz) ) the summation tone f1 1 (1408) and the differential tone fl 1 (704) are produced. If one uses only the frequencies that fall in the range of the harmonic series (Fig. 1), the most clearly defined Lissajous figures seem to be produced. The sums and differences of the harmonic 1 series are reinforced. This can be seen in Fig. 1 where fl (1408) is found at x= 8, y=2 and fl1 (704) at x= 8, y=4. That the changes in shapes of the figures are due to the beats of the tones, indicating that beats are an acoustical rather than a psychological phenomenon [6], should be investigated further. Most present-day synthesizers employ equal temperament, the division of the octave into 12 equal parts (semitones). The natural harmonic series (just intonation) is based on pure overtones-that is, if u is the frequency of a tone, then 2u is the octave (first overtone), 3/2 u is the interval of the fifth and 5/4 u is the interval of the major third, none of which (except for the octave) corresponds precisely to their equivalents in the equal temperament system. The Pythagorean tones within an octave could number as few as nine tones or as many as one could conceive in a microtonal context, depending on how far the table is enlarged beyond the audible 16 overtones and undertones. 4. Lissajous figures are made up of continuous closed paths. The closed paths are produced when any two frequencies are frequencies of the harmonic series. When two sounds are almost the same frequency, the pattern is elliptical. (Two tones of the same frequency produce a closed circle.) The slope of the major axis of the ellipse is related to the phase shift between the two tones. The two tones fl and fP1, an octave apart, produce a figure-eight pattern. A high frequency tone combined with one of a lower octave produces a multiplicity of lines in the figure-eight

C ~c C>C A C C> C C> X n A A A A AO C?*al~~~~ C AfAp C AO A A A' A C C> C> C CC C APA A'A> A?A At A A A A C; A C C A C C C C C A I B E1'' F'F" F F" EEl EE FF F F F EEb EF E,^F IC HARMON FIFTH HARMONIC * *- * * * * *l * U * U * U* U * * *U *U * U * * *m* * * * * * * * * *.* * ** * U U * * * * * U * U * * U * * * Eb7 X n SEVENTH U * * * _* U * U * * * HARMONIC _ * * U U *-U-U-----m F F F F F E< F F E< F F F E< F Etb IF~F EFCF F F F F F F 2Xn U * * * * U U * * ** ** * * * * * *s U F Fig. 4. Key for making the tone patterns shown on Fig. 3 (cf. Fig. 1). U U * F,F El 7 X n * * EE; *U F F Ec< :* F F Et E< ES~ E< E< U * F F , Et A 5 X n 3Xn F F F E< E Eb C A F F F F Eb' THIRD A Li'V EF Fig. 2. Untitled.felt pen on canvas, 23 x 68 cm, 1977. A' At A Cl 2Xn FIRST HARMONIC & SECOND HARMON IC Fig 3. Four tone patterns based on the location in the array (Fig. 4) of (1) thefirst harmonicand the second harmonic;(2) the third harmonic; (3) the fifth harmonic and (4) the seventh harmonic. overtones fl 1, fl 1, fl and f and its undertones f, f1, f1 and f 11. The resulting pattern of black squares I then use to generate a drawing consisting of four quadrants in which quadrant I of the drawing is the mirror image of the tone pattern shown in Fig. 3 (bottom right). Then the drawing is completed by producing in quadrants II and III the mirror image of the quadrant pair I and IV. One can make drawings in a similar way using the other three tone patterns in Fig. 3. If the Lambdoma diagram in Fig. 1 were continued beyond the 16th harmonic series to higher and lower frequencies, it would be found that the black square patterns would be repeated. This repetition can be detected in the picture reproduced in Fig. 5 (cf. color plate), where I show the diagram extended to the 177th overtone and the 90th undertone of the harmonic series. The colors employed correspond to an arbitrary color code that I established for certain tones [1]. References pattern resembling a projection of the Mobius strip. A fifth combined with a fundamental tone can produce crown-like patterns. A low, barely audible tone, around 20 Hz, coupled with a harmonic tone produces welldefined figures. The Lissajous figures serve me as a source of inspiration. Figure 2 shows a drawing containing shapes based on Lissajous figures and tone patterns, with bands of colored squares, suggesting the Lambdoma array. Four patterns are shown in Fig. 3; these are derived from the array shown in Fig. 1, as indicated in Fig. 4. The pattern for the first harmonic (Fig. 3, bottom right), for example, was obtained as follows: I blackened all the squares in Fig. 1 occupied by the fundamental fl and its 1. B. Hero, Paintings Based on Relative Pitch in Music, Leonardo 8, 13 (1975). 2. B. Hero, Eyes+ Ears= Ideas (New York: Franklin Furnace, 1975). 3. P. G. Brosens and E. Grenda, Applications of Galvanometers to Laser Scanning, paper presented at the 18th Ann. Tech. Mtg. of the Soc. of Photo-Optical Instrumentation Engrs., San Diego, Calif., Aug. 1974. 4. G. Trythall, Principlesand Practice of ElectronicMusic (New York: Grosset & Dunlap, 1973). 5. S. R. Wagler, Sonavision: A Visual Display of Sound, Leonardo 3, 443 (1970). Also in Kinetic Art: Theory and Practice, F. J. Malina, ed. (New York: Dover, 1974) p. 162. 6. W. Apel, Harvard Dictionary of Music, 2nd rev. ed. (Cambridge, Mass.: Harvard Univ. Press, 1969).

Top left: Barbara Hero. Untitled,felt pen on canvas, 90 x 140 cm, 1975. (Fig. 5, cf. page 303) Top right: Naomi Boretz. 'Basedon Yellow',watercoloron paper, celluloseacetatefilm, 42.5 x 54.0 cm, 1973. (Fig. 3, cf. page 294) Center left: P. P. Konchalovsky.'Portraitof a Fiddler',oil on canvas, 143 x 105cm, 1918. (Fig. 6, cf. page 321) Center right: Michele Andal. Untitled, black ink and acrylic paint on paper, 91 x 56 cm, 1977. (Fig. 8, cf. page 272) Bottom left: Tom Kelly. 'Marilyn Monroe', color photograph. (Copyright C Tom Kelly, 1978) (Fig. 6, cf. page 316) Bottom right: Anait A. Stephens. 'Eye', cylindrical white-light transmissionhologram (120?), dia. 40 cm, height 26 cm, 1975. (Fig. 2, cf. page 306) [facing page 292]