Arepulsive monster, a woundedtail-lashing serpent, dealing wild

and furious blows as it stiffens into itsdeath agony.” These were the com-ments of a critic after a performanceof Beethoven’s second symphony inthe early 19th century. About Bartok’scompositions, a critic wrote: “Somecan be played better with the elbows,others with the flat of the hand. Nonerequire fingers to perform or ears tolisten to.” On August 1, 1919, TheMusical Times in London said ofStrawinsky’s Sacre du Printemps: “Themusic of Le Sacre du Printemps baf-fles verbal description.... Practically ithas no relation to music at all as mostof us understand the word...” Thesecomments exemplify the fundamentalproblem underlying judgement andanalysis of music, namely the absenceof a clear objective criterion thatwould tell “right” from “wrong.” This isin contrast to genuinely quantitativesciences, such as physics or chemistry,where it may be assumed that a trueanswer exists and can be found byrepeated experiments and appropriatetheoretical modeling. In music, thereis usually no clear definition of opti-mality or, if there is, no unique optimal

answer exists. Moreover, not all ques-tions in musicology can be answeredby repeated experiments. Consider,for instance, the task of finding musi-cal structures in composed music.Traditional musical analysis startswith historic information that helps usto focus on structures that are likely tobe present. For example, if we analyze

a classical sonata, a harmonic analysisor an analysis of motifs is based on thewell-defined form of a sonata. Moredifficult, but often also more interest-ing, is the question whether there is

Which aspects of music can be described by quantitativemodels?

Music - Chaos, Fractals, andInformation

Jan Beran

“

Wolfgang Amadeus Mozart

Ludwig van Beethoven

Johan Sebastian Bach

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structure beyond the standard rules. Itis, however, not clear a priori whichstructures that may be discovered bydata analytical methods are musicallyimportant. For instance, one may askwhy Beethoven’s symphonies arecoherent, how he sustains the sus-pense over long stretches of time andhow this should be reflected by a per-formance. With respect to perfor-mance, repeated experiments can behelpful for finding similarities and dif-ferences between different styles ofperformance. The question of howthese are related to structures in thescore is much more difficult. Anappropriate analysis of the score isneeded, and that is where the ques-tion of identification and quantifica-tion of relevant structures is mostchallenging.

The lack of precise definitions andobjective functions, together with theemotional effect of music, have led toa tradition of purely qualitative, most-ly descriptive, and occasionally ratheremotional music reviews and a pre-dominance of historic aspects inmusic theory. Nevertheless, through-out the centuries there were occasion-al attempts to gain a more quantitativeunderstanding of music. The most

obvious quantitative approach is dueto the physical nature of performedmusic as a sound wave. For instance,the Pythagoreans in ancient Greek(fifth century B.C.) were aware of themusical significance, and apparentlypleasant effect, of simple frequencyratios such as 2/1 (octave), 3/2 (fifth),4/3 (fourth) etc. A systematic physicalunderstanding of musical sounds andacoustics was initiated by path break-ing contributions of the Germanphysicist Helmholtz. Musicalacoustics and sound engineering isnow a well-developed scientific disci-pline (see e.g. Bailhache 2001 for ahistoric account on musical acoustics)with an abundance of commercialapplications (music recording, com-puters, synthesizers, portable phones,digital television, computer games,etc.).

Acoustics is, however, only oneaspect of music. Music is not just anarbitrary collection of sounds, butrather “organized sound,” or, as theGerman philosopher and mathemati-cian Leibniz (1646-1716) put it:“Music is the arithmetics of the soul.”Whether we listen to a sonata, a sym-phony, or an Indian raga, “logical” con-struction is an inherent part of music.

Standard musical techniques, such asretrograde, inversion, arpeggio, or aug-mentation are mathematical func-tions. In the 20th century a number ofcomposers, such as Schönberg,Webern, Bartok, Xenakis, Cage,Lutoslawsky, Eimert, Kagel,Stockhausen, Boulez, and Ligeti evenused explicit mathematical formulasand ideas for their compositions,though perhaps it was psychologicallynot too clever to admit this publicly.The general audience tends to have amore romantic view of music andwants to relax. A sober explanation ofthe logical construction is likely tostigmatize a composition as purelyintellectual. Musical Opinion(London) wrote in December 1949about Schoenberg’s Piano Pieces op.11: “The Three Pieces, op.11, are nowmore than 40 years old; they are rarelyperformed, which is not surprising,since no pianist of the first rank wouldbother to learn, or desire to inflict onhis audience, such unrelievedly uglyand unrewarding abstractions...Schoenberg states: ‘I write what I feelin my heart.’ If this is really so, we canonly assume that from 1908 or so,Schoenberg has been suffering fromsome unclassifiable and peculiarly vir-

German philosopher andmathematician Leibniz

(1646-1716) put it:

“Music is thearithmetics of

the soul.”

8 VOL. 17, NO. 4, 2004

Gottfried Wilhelm Leibniz

ulent form of cardiac disease.” Also inthe 20th century, a number of theoret-ical attempts were made to developgeneral mathematical foundations ofmusic. In particular, in the last twodecades, a number of modern mathe-matical and statistical methods havebeen developed (Mazzola 2002, Beran2003).

So, how much can music be under-stood in a quantitative manner?Completely or not at all? The truthprobably lies somewhere in between:Some aspects of music may bedescribed and understood by quantita-tive models, while other aspects maybe less accessible to a mathematicalapproach. Here, and in the subse-quent articles of this volume, some(but not all) of those aspects are dis-cussed where quantitative investiga-tions are possible.

1/f-noise, Fractals, and Chaos

In 1975, Voss and Clarke formulated aprovocative statement that may besummarized – in simplified form – asfollows: Music is 1/f-noise. Their con-clusion is based on spectral timeseries analysis of recorded music.After eliminating high frequencies bylow-pass filters, the frequency struc-ture was analyzed by looking at theempirical spectrum (periodogram).The results indicated a pattern thatseemed to be common to recordedmusic in general: The value of thespectrum increases with decreasingfrequency f, and the increase is pro-portional to 1/f (hence the name 1/f-noise).

There are at least two reasons whythis statement led and still leads tocontroversial discussions among musi-cologists. Firstly, 1/f-noise-processesare purely random. In contrast, musi-cal compositions are highly organizedand presumably deterministic (exceptfor aleatoric music). To associatemusic with a purely random object istherefore rather disturbing. A justifica-tion can be given as follows: Fitting astochastic process can provide adescriptive summary of some essentialfeatures, even though the structuremay be deterministic. For instance, ifthe same pattern were repeated exact-

ly with frequency f=0.1 Hz (every 10seconds), then a spectral analysiswould reveal exactly this property inthe form of a distinct peak at this fre-quency. More generally, a spectralanalysis reveals how much of theobserved signal is due to periodicitieswith specific frequencies. If repeti-tions are only approximate, then thepeak in the spectrum is less pro-nounced. Now, in real music, patternsare rarely repeated exactly. Instead,variations are applied to both, the pat-terns (e.g., melodies) and their fre-quencies. The empirical spectrum ofsuch a disturbed periodicity then mayresemble the empirical spectrum of arandom process with similar proper-ties. It can therefore be characterizedby the spectrum of a matching ran-dom process. In this sense, it is mean-ingful to associate a recorded musicalsignal, or other musical data, with arandom process. In other words, therandom process (or its spectrum)summarizes the degree of variationand memory in a musical signal.

The second thought-provokingstatement by Voss and Clarke is thatall (low-pass filtered) music has the

same spectrum, namely proportional to1/f. This statement may indeed be toogeneral. To investigate this, let us askthe following question first: Whichaspects of a composition does recordedmusic represent? The sound wave of amusical performance is mainly deter-mined by the notes that are played, theinstruments, the way the instrumentsare played, and specific acoustic condi-tions (room, microphone, etc.).Musical instruments do not generatestrictly periodic signals. The attractiona Steinway piano or a Stradivari violinis due to a complex sound wave thatchanges in time. As it turns out, eventhe sound wave of a single note maylook like 1/f-noise. The reason is thatirregular slow changes imply a highcontribution of low-frequency compo-nents in the signal, which in turnimplies higher values of the spectrumat low frequencies. Figure 1 illustratesthis effect for a harpsichord sound(one “e” is played). Plotting the loga-rithm of the empirical spectrum versusthe logarithmic frequency shows, forsmall frequencies, a negative slopeclose to -1, which corresponds to 1/f-noise. This is the case, even though a

Figure 1. Harpsichord sound wave, squared sound wave (power) on logarithmic scale,together with their aggregations and spectra.

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Figure 3. Fitted values of power (–α) of spectrum near origin, categorized accordingto date of birth of composer. The results are based on 60 compositions.

trend function was subtracted before-hand to take care of the most obviousslow changes (Figure 1e). Now, onesingle note can hardly be called a com-position! So, finding 1/f-noise likebehavior in recorded music could bedue to the instrument(s) rather thanthe composer.

In order to separate the abstractcontent from instrumental sounds, thenotes in a score need to be analyzed bythemselves. This is quite easy formonophonic music, as long as theduration of notes is ignored. Focusingon onset time and pitch only, notescan be coded as pairs of positive inte-gers (x,y) with x=onset time andy=pitch. In the well-tempered tuning,an increase of pitch by one unit corre-sponds to an increase in frequency bythe 12th root of two. A difficulty arisesin polyphonic music. If two or morenotes occur at the same onset time,how do the notes add up? In recordedmusic, a natural superposition of notesis created automatically, since the dif-ferent sound waves are added. Forabstract notes, usual addition does notmake sense. For instance, adding twonotes y=1 and y2=8 (y1 and y2 are afifth apart) yields 9, which would behigher than each of them individually.Also, the average of 4.5 is meaninglessand does not correspond to any note.Brillinger and Irizzary (1998) proposeto use artificially created sound wavesbased on cosines multiplied by enve-lope functions. Essentially, this bringsus back to recorded music wheresound waves can be added withoutany problem. Results, however, againdepend on the particular choice ofenvelope functions. An advantage overrecorded sound waves is that the enve-lope functions can be designed suchthat their contribution to the low fre-quency behavior of the sound wave isnegligible. An alternative approachthat completely avoids confusionbetween score and instrumentalsound, while avoiding the problem ofsuperposition, is to define a simplifiedscore. Beran (2003) considers the so-called upper and lower envelope,defined by the sequence of highest orlowest notes, respectively. Here, welook instead at an arpeggio simplifica-tion. This means that notes occurring

Figure 2. Spectra (both coordinates logarithmic) for compositions by Bach, Scarlatti,Haydn, Mozart, Chopin, Brahms, Rachmaninoff, Prokoffieff, and Beran.

10 VOL. 17, NO. 4, 2004

not interesting and therefore a spectralanalysis shows random behavior.However, the amount of variation hasto be limited so that we are still able tonotice certain patterns and see con-nections between different parts. Wetherefore do not observe completelyindependent random events — thiswould correspond to 1/fo-noise.Instead, even patterns that are farapart have a strong similarity — thiscorresponds to 1/fα with a positivevalue of α. The amazing finding is thatthe degree of variation chosen by mostcomposers coincides with α=1, whichis at the border between stationarityand nonstationarity. For α < 1, 1/fα-noise is stationary, which implies acertain degree of stability. The pre-ferred degree of variation and long-term memory in music appears to beexactly at or close to the border α=1,beyond which the process wouldbecome unstable, in the sense that theprobability distribution would be dif-ferent at each time point. Similarinvestigations can be carried out forother aspects of a score. For instance,Figure 4 shows spectra for time seriesderived from onset-time gaps betweenoccurrences of the most frequentpitch. All three spectra in Figure 4exhibit 1/f-shape near the origin.

“...How anyone has the nerve tocall that kind of stuff music I just donot know. None of it had any tune andmost of it could equally well have beenour fat old tabby cat jumping on andoff the piano chasing after a mar-malade jar cover...” In spite of thiscomment by a distressed music lover(in a letter to the editor of MusicalOpinion in January 1961) who com-plained about Schönberg’s pianomusic, Schönberg’s music is in manyways a natural continuation of roman-tic music of the late 19th centuryrather than its destruction. In particu-lar, he seems to have used the sameamount of variation in terms of 1/f-behavior. For instance, the empiricalspectrum of Schönberg’s piano pieceop. 19, No. 2 in Figure 5 is very closeto 1/f-noise.

On a qualitative level, the connec-tion between music and 1/fα-noise mayalso appear plausible via a possible con-nection with the mathematical defini-

at the same onset time are replaced bya sequence of the same notes playedone after the other, ordered accordingto pitch (lowest first, highest last).Moreover, the exact spacing of onsettimes is ignored. The question that weaddress now is: Does the sequence ofnotes resemble 1/f-noise?

In a first step, high frequencies areeliminated by aggregation, i.e., bydividing the time axis into blocks of kobservations and taking averages overeach block. (see e.g. Beran and Ocker2001, Tsai and Chan 2004). Observedperiodograms of the aggregated seriesand fitted spectra are shown in Figure2 for the following compositions: (a)J.S. Bach: Prelude and Fugue No. 4from “Das wohltemperierte Klavier”;(b) D. Scarlatti: Sonata K381; (c) J.Haydn: Sonata op. 31 (1st Mov.); (d)W.A. Mozart: Sonata KV 333 (2ndMov.); (e) F. Chopin: Nocturne op. 32,No. 1; (f) J. Brahms: Hungarian DanceNo. 3; (g) S. Rachmaninoff: Préludeop. 23, No. 2; (h) S. Prokoffieff: Visionfugitive No. 14; (i) J. Beran: Piano con-cert No. 2 - Santi (2nd Mov.) (Beran2000). The fitted spectral densities arebased on so-called SEMIFAR-models.In the SEMIFAR-approach, an overallestimated trend function is removed

before the periodogram is calculated.All plots in Figure 2 exhibit negativeslopes, but not all of them are close to-1. This is confirmed by an extendedanalysis of 60 pieces by W. Byrd, J.S.Bach, D. Scarlatti, F. Couperin, J.-P.Rameau, L. van Beethoven, M.Clementi, J. Haydn, W.A. Mozart, Fr.Chopin, J. Brahms, C. Debussy, G.Fauré, S. Rachmaninoff, S. Prokoffieffund J. Beran, with estimated slopesvarying between -0.08 to -1.44. Thus,not all compositions resemble 1/fnoise. One may say, however, thatalmost all compositions resemble 1/fα-noise for some α > 0, and clear devia-tions from 1/f appear to be rare. Figure3 shows boxplots of the estimatedslopes -α, grouped according to thedate of birth of the composer.Remarkable in particular is that, in the“classical period” (Haydn, Mozart,Beethoven), the slopes are almostidentical for all compositions consid-ered here, namely, very close to 1/f-noise. It may be worth investigatinghow much this may have to do with theclassical form of a sonata. In our inves-tigation, all pieces in this time periodwere of this form.

Intuitively, the results illustrate thefact that exact repetition is musically

Figure 4. Aggregated gaps between occurrences of the most frequent note modulooctave and observed 1/f-type spectra.

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tion of chaos. First of all, common 1/fα-noise processes are fractals in the senseof Mandelbrot (1983), with the fractaldimension closely linked to α.Mathematical chaos refers to dynamicsystems that are highly sensitive to ini-tial conditions, and with (suitablydefined) trajectories that define a frac-

tal geometric object. With respect tosensitivity, music can also be calledchaotic. This is one of the reasons whyit takes so long to become a profession-al musician. Every little detail matters.A typical example is pointed out inGeorgii’s classical book on piano music.Figure 6 shows parts of a piece from

Clementi’s Gradus ad Parnassum andthe beginning of Chopin’s Etude op.10, No. 1. In both cases, the samebasic C-major chord is decomposed.but what a difference! Compared toChopin, Clementi’s chord-decomposi-tion sounds ugly. The reason for thebeautiful Chopin sound is the orderingof notes according to the sequence ofovertones of C. Also, Chopin avoids therough sound of the third in the C-majorchord at the beginning.

Finally, perhaps a note of caution isneeded at this point. The fact that mostcompositions are related to 1/f-noisedoes not mean that one can composemusic by the simple device of a random1/f-noise or fractal generator. The frac-tal character is only one of manyaspects that define a composition. It isperhaps this misunderstanding thatlead to the failure of purely “algorithmicmusic.” However, fractals can certainlyprovide an interesting starting point forcreative work. Impressive examples ofsuch music are, for instance, GyörgyLigeti’s piano etudes and his concert forpiano and orchestra.

Entropy

One may say that fractal analysis pro-vides a global measure of the coherenceof a composition. Since music may beconsidered as transmission of informa-tion from the composer/musician to thelistener, another global feature that maybe defined is the information content(or entropy) of a composition. A well-known definition of information isShannon’s entropy. For a random exper-iment, it characterizes the averageamount of information, or surprise, inthe outcome of the experiment. If arandom experiment can have a finitenumber of outcomes, then Shannon’sentropy is maximal, if all outcomes areequally likely. On the other hand, ifonly one outcome is possible, thenentropy is zero, because we know theoutcome even before carrying out theexperiment. The usefulness of entropyin music depends on whether we areable to define suitable musical quanti-ties for which probabilities or relativefrequencies can be calculated. Let usillustrate this by a simple example. Inthe well-tempered tuning, we may

Figure 6. Comparison of Clementi and Chopin: small change – great improvement.

Figure 5. Arnold Schönberg, op. 19, No. 2 - Spectrum (in log-log-coordinates) of arpeg-gio-version.

12 VOL. 17, NO. 4, 2004

identify two values of pitch as harmon-ically the same, if they differ by one orseveral octaves. Thus, we code pitch asan integer between 0 and 11. In alge-braic terms this means that we consid-er integers modulo 12. For a given com-position, we now count for each of the12 categories how many times the cor-responding pitch occurred, and calcu-late the entropy of the resulting distrib-ution. This was done for 147 composi-tions, with dates ranging from the 13thto the 20th century. The composers are:Anonymus (dates of birth between1200 and 1500), Halle (1240-1287),Ockeghem (1425-1495), Arcadelt(1505-1568), Palestrina (1525-1594),Byrd (1543-1623), Dowland (1562-1626), Hassler (1564-1612), Schein(1586-1630), Purcell (1659-1695), D.Scarlatti (1660-1725), F. Couperin(1668-1733), Croft (1678-1727),Rameau (1683-1764), J.S. Bach(1685-1750), Campion (1686-1748),Haydn (1732-1809), Clementi (1752-1832), W.A. Mozart (1756-1791),Beethoven (1770-1827), Chopin(1810-1849), Schumann (1810-1856), Wagner (1813-1883), Brahms(1833-1897), Faure (1845-1924),Debussy (1862-1918), Scriabin(1872-1915), Rachmaninoff (1873-1943), Schoenberg (1874-1951),Bartok (1881-1945), Webern (1883-1945), Prokoffieff (1891-1953),Messiaen (1908-1992) and Takemitsu(1930-1996). Plotting entropy againstthe date of birth (Figure 7) shows a sur-prising dependence. After about 1400,entropy seems to be rising. Why this isso can be best understood by reorder-ing the 12 categories of notes. In thewestern tonal system, tonalities can beordered in a natural way according tothe circle of fourths (Figure 8).Representing the frequencies of thethus ordered categories by star plotsshows the following pattern (Figure 9):Up to the 19th century, the frequenciesare high in the circle-of-fourth neigh-borhood of a central note, whereas theyare very low for the other note cate-gories. The picture changes later, some-what dramatically (e.g., Bartok), start-ing with Scriabin, who was one of thepioneers of atonal music. Due to thereplacement of the tonal system byother principles, the circle of fourths

lost its central role. The distribution ofnotes became less predetermined sothat entropy increased. An even clearertemporal development can be seenwhen looking at the ratio of the entropyof frequencies in the circle-of-fourthneighborhood of the central note and

the overall entropy (Figure 10). Theratio decreases in time, because themore a composition relies on the circleof fourths, the more uniform the distri-bution of notes is in the neighborhoodof the central note. Thus, compared tothe overall entropy, the local entropy

Figure 8. The circle of fourths.

Figure 7. Entropy vs. date of birth.

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tends to be high. This is not the casewhen the circle of fourths plays no roleor a less prominent role.

Score Information andPerformance

An area of musicology where statisticsplays a major role is performance theo-ry. The reason is that repeated observa-tions are available and experiments canbe designed to answer specific ques-tions. For music that is played from ascore, the main question is: How doesa performer translate information givenin a score into a performance? The firstquestion is how to quantify informationcontained in a score. Beran andMazzola (1999; also see Mazzola 2002and (e.g., Beran 2003) encode structur-al information of a score by so-calledmetric, harmonic, and melodic weightsor indicators. These curves character-ize, for each onset time or even for each

Figure 10. Local entropy divided by total entropy, plotted against the date of birth ofthe composer.

Figure 9. Star plots of note frequencies, with note categories ordered according to the circle of fourths.

14 VOL. 17, NO. 4, 2004

note, its metric, harmonic and melodicimportance. A slightly modified motivicindicator that takes into account a pri-ori knowledge about important motifsin the score is defined in Beran (2003).Figure 11 shows motivic indicatorfunctions for eight different motifs inSchumann’s Träumerei. These can berelated to performance data in variousways. For instance, if the tempo of aperformance is recorded, we may applyregression techniques or data sharpen-ing. Score-related data-sharpening canbe done, for instance, by consideringtempo values only at onset times wherea structural curve exceeds a certainthreshold. For example, Figure 12shows tempo for onset times where thefirst motivic indicator exceeds its 90thquantile. The tempo curves used forthis analysis were provided to us by B.Repp. Based on the sharpened data,clear differences as well as similaritiescan be identified. The visual impres-

Figure 11. Träumerei by R. Schumann: Motivic indicators.

Figure 12. Träumerei by R. Schumann: Tempo sharpened by 90%-quantiles of main motif.

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sion that Horowitz clearly differs fromCortot, as well as an amazing consis-tency of Cortot and Horowitz through-out several decades, is confirmed by acluster analysis of the sharpened data(Figure 13). In contrast to a clusteranalysis of the original data, clusteringis associated with motivic elements ofthe score. Of course, in a score such asSchumann’s Träumerei, several motivicstreams are present simultaneously sothat a unique causal attribution oftempo variations to one and only onemotif is not possible. Nevertheless, thestatistical analysis provides the meansfor finding possible explanations why aperformer may slow down or accelerateat specific points. For instance, in amore complex analysis involving met-ric, harmonic, and melodic curvessimultaneously, Beran and Mazzoladerived the following approximate rulefor Träumerei (e.g., Beran 2003):Tempo decreases at onset times thatare important from the point of view ofharmony and melody, whereas it tendsto increase for metrically importantpoints.

Remarks

Music is a fascinating mixture of orderand chaos. Quantitative musicologyinvolves many scientific disciplines,because music is neither a purely phys-ical nor a purely philosophical phenom-enon. Music is not easy to quantify,when quantification is aimed at gainingat a better understanding of music. Allthis makes music an interesting and atthe same time highly challenging topicfor interdisciplinary research. In partic-ular, due to its interdisciplinary nature,statistics plays an important part in thisemerging field. Some of us may per-haps fear that music could lose itscharm, once it is explained “by num-bers.” Yet, just like in other sciences,each new insight is likely to reveal morequestions than answers, thus tauntingthe curious to further explore the mys-terious nature of the universe.

Acknowledgements

I would like to thank B. Repp for pro-viding us with the tempo measure-ments.

References

Bailhache, P. (2001). Une histoire del'acoustique musicale. CNRSEditions.

Beran, J. (2000). Santi. col legno,WWE 1CD 20062 (http://www.col-legno.de).

Beran, J. (2003). Statistics inMusicology. Chapman & Hall,CRC Press, Boca Raton.

Beran, J. (1994). Statistics for Long-memory Processes. Chapman &Hall, London.

Brillinger, D. and Irizzary, R.A. (1998).An investigation of the second- andhigher-order spectra of music.Signal Processing, Vol. 65, 161-179.

Mandelbrot, B.B. (1983). The FractalGeometry of Nature. Freeman &Co., San Francisco.

Mazzola, G. (2002). The Topos ofMusic. Birkhäuser, Basel.

Voss, R.F. and Clarke, J. (1975). 1/fnoise in music and speech. Nature,Vol. 258, 317-318.

Figure 13. Träumerei by R. Schumann: Clusters of tempo curves, obtained after sharpening by 90%-quantiles of main motif.

16 VOL. 17, NO. 4, 2004