"1/f "in :Music from 1/f nOiSe RiChardF. Vossa) and JohnClarke Departmentof Physics, University of California andMaterials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, California94720 (Received30January 1976; revised 8 September 1977} Thespectral density offluctuations inthe audio power ofmany musical selections andof English speech variesapproximately asl/f (f isthe frequency) down to a frequencyof5X 10 -4 Hz.This resultAmplies thatthe audio-power fluctuations arecorrelated over all times in thesame manner as"l/f noise"in electroniccomponents. Thefrequency fluctuations ofmusic also have a 1/f spectral density atfrequencies down tothe inverse ofthe length of the piece of music. The frequency fluctuations ofEnglish sp•, h have a quitedifferent behavior, with a singlecharacteristic timeof about 0.1 s, the average length of a syllable.The observations onmusic suggest that 1/f noiseisa goodchoice for stochastic composition. Compositionsin which the frequency andduration ofeach note were determined by1/f noisesources soundedpleasing. Those generated bywhite-noise sources sounded toorandom, while those generated by l/f 2 noisesounded too correlated. PACSnumbers: •,3.75.Wx, 43.60.Cg, 43.75.--z, 43.70.--h

INTRODUCTION A secondcharacterization of the average behavior of V(t) is the autocorrelationfunction, (V(t)V(t+ 7)). The spectral density of many physical quantities varies as 1/f •, wheref is the frequencyand 0.5• < 7 •<1.5, over (V(t)V(t+•')) is a measureOf how the fluctuating quanti- manydecades. Thusvacuum tubes, • carbonresistors, a ties at times t and t + ß are related. For a stationary semiconductingdevices, s continuous4's or discontinuous6 process(V(t)V(t + •)) is independentof t anddepends metal films, ionic solutions,? films at the supercondUct- only on the time difference 7. Sr(f) and (V(t)V(t+ are not independent, but are related by the Wiener- ing transition,6 Josephsonjunctions, 9 nerve membranes,•ø Khintchine relations •4 sunspotactivi{y, zl andthe floodlevels of the river Nilezl all exhibitwhat is knownas "1If noise."t2 Althoughthis phenomenonhas been extensively studied, there is as yet no single theory that satisfactorily explains its ori- (V(t)V(t+7)) =•: Sv(f) cos(2rfr) df (1) gin. In this paper,z3 we showthat the audiopower and and frequency fluctuations in commontypes of music also havespectral densities that vary as 1/f. The 1/f behav- ior implies some correlation in these fluctuating quanti- Sv(f)=4 J•' (V(t)¾(t +r)) cos(2r/r)dr. (2) ties over all times corresponding to the frequency range Many fluctuating quantities may be characterized by for whichthe spectral densityis 1/f. The observation of the 1/f spectraldensity in musichas implications for a singlecorrelation time re. In sucha case,V(t) is correlatedwith V(t + •) for 171<< •½, andis independent music'compositionalprocedures. We haveused a of V(t+•') for I• '>>•:. Usually, (V(t)v(t+ noise source in a simple computer algorithm to produce xexp(- I •1/•:). From Eq. (2), it is thenpossible to stochasticmusic. The results suggestthat 1/f noise showthat Sr(f) is "white" (independentof frequency)in sources have considerable promise for computer com- the frequencyrange• corresponding to times over which position. V(t) is independent(f<< 1/2•r•-:); and is a rapidly decreas- ing functionof fi•equency,usually 1/f •, in the frequency h SPECTRAL DENSITY AND TIME CORRELATION S range•ver whichV(t) is correlated(f >>1/2•r•'½). A Although frequently used in the analysis of random quantity with a 1/f spectral density cannot, therefore, be sigrmls or "noise," the spectral density (power spec- characterized by a single correlation time. In fact, the trum) is an extremely useful characterization of the 1/f spectral densityimplies some correlation in V(t) average behavior of any quantity varying in time. The over all times corresponding to the frequency range for spectral densitySv(f) of a quantity V(t) fluctuatingwith whichSt(f) is 1/f-like. •s In general,a negativeslope time t is a measure of the mean squaredvariation for Sr(f) implies some degree of correlation in V(t) over in a unit bandwidth centered on the frequency f. The times of roughly1/2•rf. A steepslope implies a higher average is taken over a time that is long compared with degree of correlation than a shallow slope. Thus a the period; in practice, we usually avdrage over at quantitywith a 1/f • spectraldensity is highlycorrelated. least 30 periods. $v(f)may be measuredby passing V(t) througha tunedfilter of frequencyf andbandwidth Figure 1 showssamples of white, l/f, .and1/f a noise õf. Sv(f) is then the average of the squaredoutput of voltages versus time. Each fluctuating voltage was am- the filter dividedby 5f. Thus, if V(t) is a voltage, Sv(f) 'ptifiedto coverthe samerange, and each had the same is in units of volt squared per hertz. high-frequency cutoff. The has the most random appearance, and shows rapid uncorrelated a•Present address: IBM Thomas J. Watson Research Center, changes.The 1/• • noiseis the mostcorrelated showing Yorktown Heights, NY 10598. only sIow changes. The l?f noise is intermediate.

258 J. Acoust.Soc. Am. 63(1), Jan. 1978 258 259 R.F. Vossand J. Clarke:l/f, noiseand music 259

V(i)• t (a) WHITE NOISE

I/f NOISE I/f2NOIS• ßFIG. 1. Samplesof white, l/f, and1/f 2 noisevoltages, V(t) versus •_me t. (b)

II. 1/f NOISE IN MUSIC

In our measurements on music and speech, the fluc- tuating quantity of interest was converted to a voltage whose spectral density was measured by an interfaced PDP-11 computer using a Fast Fourier Transform algo- rithm that simulates a bank of filters. The most famil- iar fluctuating quantity associated with music is the audiosignal V(t) suchas the voltageused to drive a

-5 -2 -I 0 I 2 3 4 LOili0(t/IHz) FIG. 3. Bach's First BrandenburgConcerto (log scales). (a) $•(f) rs/; (b)$vi(f) vsf.

speaker system. Figure 2(a) showsa plot of the spec- tral density$v(f) of the audiosignal from J. S. Bach's First Brandenburg Concerto averaged over the entire concerto. The spectral density consists of a series of sharp peaksin the frequencyrange 100Hz to 2 kHz corresponding to the individual notes in the concerto and, of course, is far from 1/f. Althoughthis spec- trum contains much useful information, our primary interest is in more slowly varying quantities. o I000 2(XX) One such quantity we call the audio power of the music, f (Hz) whichi• proportionalto the electricalpower delivered to a loudspeakerby an amplifierand hence to Vl(t). Va(t)varies monotonicallywith the loudnessof the music. In order to measureVi(t) the audiosignal was amplified and passed through a bandpassfilter in the range 100 Hz to 10 kHz. The output voltage was squared, and filtered with a 20-Hz low-pass filter. This processproduced a slowlyvarying signal, Vi(t), that was proportional to the "instantaneous" audio power of the music. Correlationsof V2(t)represent correlations of the audio power of successive notes. The spectral density of the audio power fluctuations of the First BrandenburgConcerto, S•l(f), averagedover the entire concerto is shownin Fig. 2(b). On this linear-linear plot, the .audio power fluctuations appear as a peak close to zero frequency. o 2 4 6 Figure 3 is a log-tog plot of the same spectra as in f (Hz) Fig. 2, In Fig. 3(a), the spectral density of the audio FIG. 2. Bach's First BrandenburgConcerto (linear scales). signal, S•(f), is distributedover the audiorange. In (a) Spectraldensity of audiosig•l, Sv(f) vs/• (b) spectral Fig. 3(b), however, the spectral densityof the audio densityof audiopower fluctuations,S•,2(f) vsf. powerfluctuations, $rdf), showsthe 1/f behaviorbe-

J. Acoust.Snc. Am., Vol. 63, No. 1, January1978 260 R, F. Vossand J. Clarke: 1/f noiseand music 260 low 1 Hz. The peaks between 1 and 10 Hz are due to the rhythmic structure of the music. 10 m ' Figure 4(a) showsthe spectral density of audio power fluctuations for a recording of Scott Joplin piano rags 9 - I/F averaged over the entire recording. Although this music has a more pronounced metric structure than the Bran- denburgConcerto, and, consequently, has more struc- ture in the spectral density between 1 and 10 Hz, the spectral densitybelow I Hz is still 1/f-like. ß 7 In order to measureSv2(f) at frequenciesbelow 10 '2 Hz an audio signal of greater duration than a single record is needed, for example, that from a radio sta- tiono The audio signal from an AM radio was filtered and squared. Sv,.(f) was averagedover approximately 12 h, and thus included many musical selections as well as announcementsand commercials. Figures 4(b) through(d) s•howthe spectral densitiesof the audiopow- o. er fluctuations for three radio stations characterized by different motifs. Figure 4(b) showsSv•(f) for a classi- cal station. The spectral densityexhibits a smooth1/f dependence. Figure 4(c) showsS•,2(f) for a rock sta- tion. The spectraldensity is I/f-like above2x 10's Hz, and flattens for lower frequencies, indicating that the I ) correlation of the audio power fluctuations does not ex- tend over times longer than a single selection, roughly I I I I I 100s. Figure 4(d) showsSg,.(f) for a newsand talk station, and is representative of S•,,.(f) for speech. -5 -• -I 0 I Onceagain the spectral densityis I/f-like. In Fig. 4(b) LOG10(f/1 Hz) andFig. 4(d), Sra(f) remains1/f-like downto the low- FIG. 5. Spectral density of frequencyfluctuations, Sz(f) vs f for four radio stations (a) classicah (b) andblues; (c) rock4 and (d) news and talk.

est frequencymeasured, 5x 10'4 Hz, implyingcorrela- tions over times of a least 5 min.

Another slowly varying quantity in speech and music is the "instantaneous" frequency. A convenient means of measuring the frequency is by the rate Z of zero crossings of the audio signal, V(t). Thus an audio sig- nal of low frequency will have few zero crossings per • 5 (o) second and a small Z, while a high-frequency signal will have a highZ. For the case of music, Z(t) roughly follows the . Correlations in Z(t) represent cor- relations in the frequencies of successive notes. Fig- ure 5 shows the spectral density of the rate of zero crossings, Sz(f), for four radio stations averagedover approximately12 h. Z(t) was also smoothedby a 20-Hz low-pass filter before the spectral density was mea- suredo Figure 5(a) showsSz(f) for a classical station. The spectraldensity varies closelyas 1/f above4x 10'4 Hz. Figures 5(b) and 5(c) showSz(f) for a jazz and blues station and a rock station. Here the spectral den-

I sity is 1/f-like downto frequenciescorresponding to the average selection length, and is flat at lower fre- quencies. Figure 5(d), however, which showsSz(f) for 0 I a news and talk station, exhibits a quite different spec- -3 -2 -I 0 I tral density. The spectral density is that of a quantity LOGlo(f/1Hz) characterized by two correlation times: The average FIG. 4. Spectraldensity of audiopower fiuctuatioas, Sv2(f) length of an individual speech , roughly 0.1 s, and vsf for (a) ScottJoplin piano rags; (b) classicalradio station; the average length of time for which a given announcer (c) rock station• and (d) news and talk station. talks, about 100 s. For most musical selections the

J. Acoust.Soc. Am., Vol. 63, No. 1, January1978 261 R.F. Vosmand J. Clarke:1If noiseand music 261

I I { the other pieces showdecreased correlations for times longer than about 10 s.

The I/f behaviorof quantitiesassociated with music and speech is, perhaps, not so surprising. We specu- late that measures of "intelligent" behavior should show a 1/f-like spectral density. Whereas a quantitiywith a - (hi white spectral density is uncorrelated with its past, and a quantitywith a 1If •' spectraldensity depends very _ (c) stronglyon its past, a quantitywith a 1/f spectralden- sity has an intermediate behavior, with some correla- tion over all times, yet not dependingtoo strongly on its past. Human communication is one example where cor- - (d) _ relations extend over various time scales. In music much of the communication is conveyed by frequency changesthat exhibita 1If spectraldensity. In English speech, on the other hand, the communicationis not directly related to the frequencies of the individual : Successive sounds may convey related ideas even thoughtheir frequencies are statistically uncorre- lated. In other words, the ideas communicated may have long time correlations even thoughthe frequencies of successive sounds are unrelated. It would also be of interest to investigate the music and speech of other cultures, such as Chinese, in which pitch plays an im- portant role in communication. -3 -2 -I 0 LOGto(f/! FIG. 6. Audiopower fluctuation spectra densities, Sv'(.f) vs f for (a)Davidovsk•' s Synchronism I, H, andI• (b)Babbit's i I I StringQuartet number 3• (c) Jotas'(•_•,et number3• (d) Carter's Piano concerto in two movements• and (e) Slx)ck- hansen's 1V[omen[e.

frequency contenthas correlations that extend over a large rangeof timesand, consequently,has a 1/f-like spectral density. For normal English speech, on the (c) other hand, the frequencies of the individual speech sounds are statistically unrelated. As a result, the spectral density is "white" for frequencies less than about2 Hz, andfalls as 1If 2 for f>-2 Hz. In fact, in Figs. 5(a)-5(c), one observesshoulders at about 2 Hz correspondingto speechaveraged in with the music. The prominence of this shoulder increases as the vocal content of the music increases, or as the commercial interruptions become more frequent. Figures 6 and 7 show the measured audio power and frequency fluctuation spectral densities for several pieces by different composers. In each case the spec- tral density was averaged over the length of the piece in the manner described above. Although all of the pieces show an increasing spectral density at lower frequen- cies, individual differences can be observed. For the 1 I I audiopower fluctuations (Fig. 6), the selectionsby -3 -2 -I 0 Davidovsky[Fig. 6(a)]and Stockhausen [Fiõ. 6(e)]show the correlationscharacteristic of the 1/f spectralden- LOGtO(f/! sitywhile those by Babbit[Fig. 6(b)], Jolas[Fig. 6(c)], FIG. ?. Frequencyfluctuation spectral densities, Sz(f) andCarter [Fig. 6(d)]show decreasing correlations at f for (a) Davidovaky'eSynchronism l, II, andLD; (b) Babbit's times longer than several seconds. For the frequency StringQuartet number 3; (c) Jolas'Quartet number 3; (d) fluctuations(Fig. ?), Davidovsky'sSynchronism [Fig. Carter's •iano concerto in two movemeutsi and (e) Stock- ?(a)]remains closest to the 1/f spectraldensity, while hansen' Morncute.

J. Acomt Soe.Am., VoL 63, No. 1, Januaw1978 262 R.F. Vossand J. Clarke: 1/f noiseand music

III. 1/f NOISE AND STOCHASTIC COMPOSITION Theobservation of{if spectraldensities for audio power and frequency fluctuations in music has implica- tions for stochastic music composition. Traditionally, • I , I •_-7J. I . _.• stochastic compositions have been based on a random number generator (white noise source) which is uncor- .... -• ,-4.' "- ..... --::.-: ; - -- related in time. In the simplest case the white noise source can be used to determine the frequency and dura- tion (quantized in some standard manner) of successive •.? •. .•.. •,, .•.. , •.==•_•. !.--- '-•. notes. The resulting music is and soundsstructureless. (Figure 8 shows an example of this "white music" which we have producedusing a white noise source.) Most work on stochastic composition has been concerned with 1/! MUSIC ways of adding the time correlations that the random FTG. 9. •'requellcyand duration determined by a lff noise numbergenerator could not provide. Low-level Markov source. processes(in which the probability of a given note de- pendson its immediatepredecessors) were ableto im- voltageacross a resistor. The 1if noisewas obtained pose some local structure but lacked long time correla- from the voltage fluctuations across a current-biased tions. Attempts at increasing the number of preceding transistor. The 1if z noisewas obtainedby filtering notes on which the given note depended gave increasing- the white noise source by a 6 dB per octave low-pass ly repetitious results rather than interesting long term filter where the cutoff frequency was less than the in- structure.ts By addingrejection rules for the random verse of the length of the piece of music. The noise choices (a trial note is rejected if it violates one of the voltage was sampled, digitized, and stored in a PDP-11 rules), Hiller and Isaacson were also able to obtain computer as a series of numbers whose spectral density local structurebut no longterm correlations? J. C. was the same as that of the noise source. These num- Tenney has developed an algorithm that introduces long- bers were rounded and scaled to represent the notes of term structure by slowly varying the distribution of ran- a standard musical scale (pentatonic, major, or 12 dom numbers from which the notes were selected. t8 tone chromatic) over a two-octave range: A high num- Thus, although it has been possible to impose some ber specified a high frequency and vice versa. This structure on a specific time scale, the stochastic music process was then repeated with another noise source to has been unable to match the correlations and structure produce an independent series of stored numbers whose found in music over a wide range of times. values corresponded to the durations of successive notes. We propose that the natural means of adding this The PDP-11 was then used to "perform" the stochas- structureis with the use of a 1if noisesource rather tic composition by controlling a single amplitude modu- than by imposing constraints upon on white noise source. lated voltage controlled oscillator. We used sinusoidal, The 1if noisesource itself has the sametime correla- square, and triangle waveforms and a variety of attack tions as we have measured in various types of music. and decay rates. The computer was also used to put the To illustrate this process at an elementary level, we stochastic compositions in more conventional form. presentshort typical selectionscomposed by white, l/f, Samples of these computer "scores" are shown in Figs. and1if 2 noise. 8-10. Accidentals apply only to the notes they precede. In each case a physical noise source was used to pro- The scores are presented without bars since no con- duce a fluctuating voltage with the desired spectrum. straints were imposed on durations of successive notes. The white noise was obtained from the Johnson noise In Fig. 8 a white noise source was used to determine frequencyand duration. In Fig. 9 a 1if noise source wasused, while in Fig. 10 a 1if • noisesource was used. Although Figs. 8-10 are not intended as complete for- mal compositions, they are representative of the cor- relations between successive notes that can be achieved when the three types of noise sources of Fig. I are used to control various musical parameters. In each case the noise sources were "Gaussian" implying that values near the mean were more likely than extreme values. Over a period of about two years we have played 'sam- ples of our music to several hundred people at nine uni- versities and research laboratories. The listeners ranged from those with little technical knowledge of music to professional musicians and composers. We playedselections of White, i/f, and1/f • musicvarying WHITE MUSIC in lengthfrom oneto ten minutes. Our 1/f musicwas FIG. Frequency and duration determined by a white noise judged by most listeners to be far more interesting than either the white music (which was "too random'•) or the

J. Acou•. Soc.Am., Vol. 63, No. 1, January1978 263 R.F. Vo. andJ. Clarke:1/f noi•eand music 263

trol various synthesizerinputs providing correlated but random variations.

ACKNOWLEDGMENTS

We are pleased to acknowledgehelpful conversations with and encouragement from Dr. M. Mathews and A. Moorer. We are particularly indebted to Edwin E. Dugger for his advice, comments, and a critical reading of the manuscript. Finally, we are grateful to Dr. Robert W. Young for his helpful comments during the revision of the manuscript.

1j. B. Johnson,Phys. Rev. 26, 71-85 (1925). 2C.J. Christensonand G. L. Pearson,Bell Syst. Tech.. J. 15, scalelike1/f 2 music(which was "toocorrelated"). In- 197-223 (1936). •For a review,see A. vander Ziel, Noise: Sources,Charac- deedthe surprisingsophistication of the 1If music te•-izatio., Mcasuzement (Prentice-Hall, EnglewoodCliffs, (whichwas close to being"just right") suggeststhat the NJ., 1970). 1/f noisesource is an excellentmethod for addingtime iF. N. Hoogeand A.M. H. Hoppeubrouwers,Physica 4õ, correlations. '386--392 (1969). r•t.F. Vossand J. Clarke,'Phys. Rev. B13, 556-573 (1976). IV. DISCUSSION 6joL. Williamsand R. K. Burdeft,J. Phys.Chem. 2, 298- '307 (1969). There is, however,more to musicthan 1/f noise: ?F. N. Hooge,Phys. Lett. 33A, 169-170(1970). , for example, represents a periodic constraint •J. Clarkeand T. Y. Hsiang,Phys. Rev.. lmlS, 4790--4800 on the duration of successive notes. Although our sim- (1976). ple algorithmswere sufficientto demonstratethat 1/f sj. Clarkeand G. Hawkins,Phys. Rev. B14,2826-2831 (1976). noise is a better choice than white noise for stochastic løA.A. Verveenand H. E. Derkson,Proc. LEEE 86, 906-- 916 (1968). composition, the variation of only two parameters (fre- liB. B. Mandelbrotand J. R. Wallis,Water Resour. Res. õ, quency and duration of the notes of a single voice) can, 321-340 (1969). at best, produce bnly a very simple form of music. 12Thespeetral density of voltagefluctuations has been measured More structure is needed, not all of which can be pro- downto 5x10'I Hz in semieonductors[M. A. Caloyannides, videdby 1If noisesources. We improvedon theseele- J. Appl. Phys. 4õ, 307-316 (1974)]• in the ease of the river mentary compositions by using two voices that were Nile, the spectral densityof the annualflood levels extends downto 3x10 '11 Hz. either independentor partially correlated (noteshaving I•Apreliminary account ofthese measurements appeared in the same duration but independent frequencies or vice Nature, 2õ8, 317-318 (1975). versa), andby varying the overall intensitywith an addi- 14F.Reft, Fu.damentals ofStatistical amt The•nal t•hysics, tional 1If noisesource. We addedmore structureto (McGraw-Hill, New ¾orl• 1965), pp. 585-587. the musicby introducingeither a Simple, constant 15Thefact that 1/f noisecontains a distribuUon of correlation rhythm, or a variable rhythm determined by another time• is discussed by Aldert van der Ziel, Physica 16, 359- 1If noisesource. We suggestthe useof 1If noise 372 (1950). sources on various structural levels (from the charac- iaSee,for example,R. C. Pinkerton,Sci. Am. 194, 77-86 (1956) or H. F. Olson and H. Belar, J. Acoust. Soc. Am. terization of individual notes to that of entire move- $8, 1163--1170 (1961). ments) coupledwith external constraints (for example, iVL. A. Hiller, Jr. andL. M. Isaacson,Ex•½7'•me. tal Music rhythm or the rejection rules of Hiller) as offering (McGraw-Hill, New York, 1959). promising possibilities for stochastic composition. A 18DiscussedbyJ. R. Pierce, M. V. Mathews,and J. C. furtherpossibility is the use of 1If noisesources to con- Risset, GravesanerBl•/tter 27/28, 92-97 (1965).

J. Acou•t. Soc.Am., VoL 63, No. 1, January1978