Article: A Bibliography of Statistical Applications in Musicology, by Nigel Nettheim

Abstract Statistical applications in musicology appear in widely scattered publications. The present bibliography, mainly of English language publications, extends back to the beginning of the present century. The analysis of musical scores is emphasized, but applications in the social sciences are also touched upon, as well as those to performance studies and algorithmic composition. Statistical techniques include simple summarization, graphical methods, time series analysis, information theory, Zipf's law, Markov chains, fractals and neural networks. Several cases of misapplication of statistics are noted. Commentary is provided on the field and its sub-fields.

Many musicologists may be unfamiliar with the possible applications of statistics to musicology; the present bibliography may serve to draw some of the possibilities to their attention, emphasizing applications to music analysis. Although a good knowledge of statistics is generally desirable for intending workers in this field, it is hoped that this presentation will be readable by all. Occasional technical statistical terms may be unfamiliar to the reader, who may then wish to consult a dictionary such as Marriott (1990). Conversely, it is hoped to interest statisticians having perhaps relatively little musical knowledge. The survey may be of assistance even for researchers already working with such applications, for the literature is widely scattered. The only previous general survey known to the present writer is the short article by Waugh (1985). The Figures are explained as far as possible in short captions; for further details the reader may consult the relevant references (Figs. 2-9).

Statistics deals largely with replication, whereas music deals with particular cases which are often works of art, and these points of view may well seem contradictory. A particular case, such as an outlier, may occasionally interest a statistician who is otherwise concerned with general tendencies, but in music particular cases are everything. Changing a single note by the smallest amount (say C to C sharp) may have little statistical but enormous musical effect, for by its nature a musical masterpiece is an organic whole, not just a series of note-decisions. On the face of it, therefore, statistics and music are not likely to mix well.

Still, some areas where a statistical view of music seems warranted have arisen. For example, there are cases where a given composer acquires a musical language whose detailed functioning remains in effect over many years and through many compositions; this stable material may lend itself to statistical investigation. Some factors may apply to a whole repertoire of music, such as the folksongs of a given nationality. Principles of melody, harmony or rhythm underlying music of the 'period of common practice' in Western art music (c.1600 - c.1900) might also be investigated statistically. Thus sufficient homogeneity of the material studied may lead to the admission of a statistical approach; this question was discussed by M. Ellis (1981) and by W. Winkler (1971: 412):

What features of a work of art may be characteristic enough for the time or place ... or the artist himself can be decided only by competent [musical] experts. If they have enough knowledge of statistical theory, they may do the work alone, otherwise in collaboration with statistical experts.

In connection with collaboration, it will be noticed that a number of the references given below have joint authors. A general exhortation for the use of statistics in musicology was given by Meyer (1989: 64):

Since all classification and all generalization about stylistic traits are based on some estimate of relative frequency, statistics are inescapable. This being so, it seems prudent to gather, analyze, and interpret statistical data according to some coherent, even systematic, plan. ... it would appear desirable to define as rigorously as possible what is to count as a given trait, to gather data about such traits systematically, and to collate and analyze it consistently and scrupulously—in short, to employ the highly refined methods and theories developed in the discipline of mathematical statistics and sampling theory.

Music may for the present purpose be divided into three areas: composition, performance and musicology; the latter may be divided, again imperfectly, into historical musicology, ethnomusicolgy and analytical musicology. The last category is the one emphasized in this paper. However, as analysis can assist the other branches of musicology as well as composition and performance, those areas will be touched upon too.

The boundaries separating the present work from several neighbouring fields can, as usual, not be drawn precisely; in particular, statistical methods naturally overlap with computer methods, for which see the surveys by Davis (1988) and Hewlett and Selfridge-Field (1985-96). Mathematical rather than statistical work, referring for example to tuning systems, to the combinatorics of musical chords, or to mathematical aesthetics, will not be covered here; nor will the statistics of sound itself, as for instance the problem of the compression of timbral spectra for the digital encoding of sound waves. The converse application, that of music or sound to statistics, or 'sonification' as an alternative to 'visualization' of data, shows some promise (Pollack and Ficks 1954; Yeung 1980; Wilson 1982; Iverson 1992).

Statistical methods may be divided, once more roughly, into descriptive, exploratory and confirmatory. It turns out that most of the musical applications to date are primarily descriptive; that is the normal beginning in any field, and this field must be considered as yet relatively little developed. However, the other two categories will be emphasized when possible.

I have attempted to cover the subject from its early history to recent developments and to include work of all standards from questionable to excellent, at all levels from elementary to advanced. While complete coverage is of course impossible, a survey of fair scope can thus be hoped for. On the other hand, few non-English-language references, articles from conference proceedings, or dissertations are included. An expert in any one sub-field may find this survey inadequate, as I have aimed not for depth so much as for breadth and accessibility and to give possible starting points for investigation.

Music analysis may be carried out for musicological purposes alone, or as a preliminary step towards subsequent synthesis, that is, composition. This section will cover work emphasizing the former, section 2.2 the latter.

The material of music analysis normally consists of the notes of musical scores, encoded in some way. The amount of relevant structure in scores is generally too great for standard statistical methods, and statistical studies have so far most often been confined to melody, understood here as a succession of single notes, and possibly rests, over time.

In a graphical representation time is normally represented as running horizontally from left to right. On the vertical axis one plots the pitch of the melody's notes. Even this simple description conceals some difficulties, however. Pitch, the result of human processing of the cyclical frequency of a note's sound-wave, obeys an approximately logarithmic law; most often, therefore, the vertical scale is taken as the logarithm of cyclical frequency, thus approximately following distances across a piano keyboard or vertically on a musical score. (A refinement, often desirable but not always made, reflects enharmonic equivalence; see Gabura 1970: 251-253.) The horizontal axis is sometimes incremented equally for each successive note in order to study mere pitch-sequences irrespective of the durations, but it is more often proportional to the duration of each note. A common way to convert a musical score to statistical data is shown in Fig. 1, where notes have been represented as if they were continued into following rests or repeated notes (which is satisfatory for some but not all purposes). The graph can readily be converted to numbers for pitch {35,32,35,32,35,33,30,33,30,33,30,33,32,28} and duration {1,1,1,1,1,1,2,1,1,1,1,1,1,2}.

The earliest appearance of such graphs known to me is found in Spencer (1902: 46) who however drew no strong conclusions from his comparisons.

Time in music is not normally an undifferentiated extension; rather, it possesses structure according to the musical metre. Furthermore, notes starting at different metrical positions have different musical significance for the listener. Until the past decade few statistical studies took metre into account.

Other graphical representations might conceivably facilitate the grasping of relevant musical relationships. Stephenson (1988) used lines and angles in four modes: (i) pitch versus time, (ii) scale degree versus time, (iii) pitch on a cycle of fifths versus time, and (iv) pitch versus metric position; these methods, though worth the attempt, do not appear to have had particular success. Tufte (1990: 59, 117) considered some possibilities relating music with concurrent dance. Brinkman and Mesiti (1991) gave many approaches, one of which is shown in Fig. 2; such approaches may yield their value not in any one example but in comparisons between works or in surveys of large repertoires for which listening would be too time-consuming and human memory inadequate, for then data compression and summary are needed.

The geometrical representation of the keys or tonalities has proven challenging, the most sophisticated attempt so far being that of Krumhansl (1990: 48) who used the surface of a toroid to represent three relationships: the cycle of fifths, the relative major/minor and tonic major/minor.

When statistics are to be collected of more detailed features of musical scores, it is important to avoid a literal-minded approach to the printed page. For instance, although most notes are printed at the same size and so are visually homogeneous, some will be more fundamental, others more decorative or subsidiary. Different metrical notations may have the same meaning, being distinguished out of purely orthographic considerations. A list of the harmonies occurring may be misleading, for some are more essential, others more transitory. The intended meaning of slurs varies with the musical situation and with the composer. The meaning of indications of tempo or character, such as adagio, may vary through the course of history. Thus the well-defined categories which statisticians naturally prefer are not always available in music, so that pre-processing by a musical expert is sometimes advisable.

A brief consideration of the databases involved in musical research may be helpful here. Folk-songs and liturgical chant have the advantage, for statistical purposes, of relative simplicity, usually being short and consisting of only a single melodic line. Hymn tunes are similar in those respects when considered without their harmonizing chords. There is, at the time of writing, no one standard code for the computer representation of even a single line of melody; some widely-used codes are discussed by Hewlett and Selfridge-Field (1985-1996). Most Western music, on the other hand, includes either counterpoint or harmony to support a melody. No computer algorithm for harmonic analysis is generally accepted, but harmonic analysis input by a human is of course suitable data for statistical analysis (e.g. Marillier 1971, 1983). The quality control even of relatively simple musical databases requires attention (Huron 1988; Nettheim 1993b).

The availability of databases to the intending researcher is an important question. Much printed music is nowadays computer-set, but the encoding scheme may be proprietory to the publisher and in any case such data is not usually available to the public. The University of Essen's large folksong database prepared by Helmut Schaffrath has been placed in the public domain, and La Trobe University's 14th-century database prepared by John Stinson is available commercially, both these being in computer-readable form. Data files in the MIDI format are currently becoming available, either in the public domain or commercially; although primarily intended for playing, these files can as well be used for analysis, but the limitations of the MIDI scheme for this purpose are considerable (Hewlett and Selfridge-Field, 1993-94, 9: 11-28). The HUMDRUM computer program for musical analysis, placed in the public domain by David Huron of the University of Waterloo, includes some musical data.

We now arrive at the paper's main purpose, a survey of applications of statistics in analytical musicological studies. The citations are generally arranged chronologically within topics.

The simplest kind of such applications involves just descriptive statistics, and the simplest of these is the counting of notes, intervals, or other patterns. Usually, of course, there is a music-analytical purpose behind the counting; mere descriptive statistics without musical understanding are unproductive. Samson (1985: 147) referred to this point in his criticism of Thomas (1963):

... she tabulates the total chord usage of Chopin's complete piano music, an astonishing feat of misdirected industry which tells us virtually nothing of significance about his style.

Similarly, Hanna (1965) undertook a heroic counting of all the notes in Schubert's piano sonatas, but convincing motivation seemed to be lacking and enharmonic equivalences were regrettably not distinguished.

In an early counting study Watt (1924) compared two widely disparate repertoires. The prominent ethnomusicologist Densmore (1972/1932) also took a basically statistical approach. Jeppeson's book (1946) describing the music of Palestrina is a classic of musicology; it is based upon the systematic counting of note-patterns (taking metre into account) and so is essentially statistical in nature, if only occasionally in appearance (see the Tables on his pp. 42-43, 284-285). Jeppeson's success was possible for such an orderly musical language but could not be expected to be reproduced for later more complex languages.

Marillier (1971, 1983) carried out one of the most thorough and musically sensitive of all studies of this type, tackling no less than the tonal progressions in every bar of all 104 symphonies of F. J. Haydn. His simple but effective graphical representation yielded instructive conclusions about Haydn's treatment of large-scale form that could not have been reached by an unaided connoisseur, no matter how expert. These conclusions concerned for example the proportions appearing in the characteristic sinusoidal behaviour of tonality in first-movement forms, that is, motion from the tonic first to the dominant and later to the subdominant sides, and finally back to the tonic. He represented the harmonic roots in his graphs according to the cycle of fifths, which is as much as can be represented linearly (cf. Krumhansl 1990: 48). He also showed the gradual increase in the scope of Haydn's tonalities during the course of his composing life, more remote flat ones gradually coming into play (Fig. 3).

In counting intervals, one is often faced with the question whether or not to include repeated notes (unisons, or zero intervals); Suchoff (1970) and Hofstetter (1979) decided not to. Morehen (1981) used several functions of his data as descriptive statistics; he explained correlation incorrectly, however: "zero [correlation] indicates a random relationship" (p. 177), whereas only the converse is true: a random relationship has zero correlation. Karlina & Detlovs (1988) counted intervals not between successive notes but between successive musical sequences. Pont (1990) restricted himself to the first two melodic directions of tunes.

Camp (1968) counted durations, investigating the appearance of the well-known 'golden mean' [ { √ 5 - 1 } / 2 = 0.618... ] in the proportions of musical large-scale forms. Her thesis was drawn out from rather slender material, more convincing studies being carried out later by Howat (1983) and Marillier (1983: 198-9).

Gjerdingen (1986, 1988) studied the appearances in composed music of the particular melodic pattern formed by the scale degrees 1-7 . . . 4-3. He thus searched for pairs of phrases of which the first ends melodically with C-B and the second answers with F-E, after transposing to C major and reducing the music by the removal of less essential notes. He showed the number of such patterns found over time, although the proportion of the phrases examined which had the indicated property would have been of interest (1986: 35). The musical basis of his work was carefully considered, but of interest here is his discussion of his graph (1986: 34) shown in Fig. 4:

One need hardly be a statistician to see that the population of the 1-7 . . . 4-3 schema approximates a normal distribution (footnote: the one apparent deviation is a slight asymmetry...)

Although Gjerdingen's purpose could have been achieved with a weaker claim than that of the normality of his distribution, his remark may be taken as something of a challenge to statisticians. Few of them would consider using the normal distribution to model a time-domain trace, and if they did so they would realise that the heavy upper tail here, although visually slight, corresponds to overwhelming unlikelhood. The use of the given pattern thus persists considerably into the future from its peak time and a Poisson distribution, for instance, might have been tried.

By referring in the quoted remark to a 'population' property, Gjerdingen raised the question of inference, and was thus faced with the problem of finding a sense in which his observations could be considered a sample. His attempts to deal with that problem (Gjerdingen 1986: 36; 1988: 262-3) are not convincing; although one sympathises with him in this difficulty, it would have been preferable to acknowledge the informality of his experiment. The question of sampling in musicology was considered also by M. Ellis (1981).

Vos and Troost (1989) showed, among other results, that in classical music and folksongs large intervals tend to ascend and small ones to descend: the natural tendency in singing is evidently to leap up and step down, rather than the converse which would require greater physical effort in total. The converse ramp effect, gradual rises followed by sudden falls, was however found to prevail for the density of musical texture (Huron, 1990a) and for loudness (Huron, 1990b). Yi (1990: 276-7) rightly emphasized the desirability of considering pitch and duration jointly, but his statistical procedures are not always convincing; for instance, his graph of a 'normal' distribution (p. 30) resembles the top half of a hamburger, whereas it should be bell-shaped. Boroda (1991) compared the rhythmic patterns appearing in the folksongs of ten nationalities. Nettheim (1993a) related pitch and duration jointly to the musical metre, suggesting an analogy to a regularly recurring pulse-like feeling in music. C. Ellis (1992) studied rhythm in an Australian aboriginal repertory; the balance between statistical detail and conciseness favoured the former, the reader therefore needing to delve deeply into the material. Gelber (1995) compared the styles of children's songs of several European nations, using extensive statistical tabulations.

Huron (1995-96) sought evidence for the predominance of the arch shape in melodic phrases of Western folksong. He used data from the Essen archive (see section 2.1.2). The paper, however, highlights some of the pitfalls awaiting the inexpert statistician. Huron did not discuss the definition of a 'phrase', instead assuming that the archive's division into 'phrases' provided appropriate raw material. However, those phrase divisions had been made by various encoders according to no specified criteria; for instance, the choice between two-bar and four-bar phrasing was in many cases made arbitrarily acccording to the personal taste of the particular encoder. Indeed it is possible that some divisions had been made, whether consciously or not, according to an assumed typical arch shape, which would introduce circularity into Huron's procedure.

Huron has also misunderstood the nature of statistical chi-square tests (as well as the spelling: not chi-squared). His statement (ibid. p.23) that "If the value of p achieves the alpha confidence level, this means that [the hypothesis] is probably true" is confused on many counts; reference to a standard statistical text would be advisable. The limitations of chi-square and other tests should also be noted. One of these is the fact that when a continuum of values is conceivable for an unknown quantity, the hypothesis that it has one particular value will be rejected by a sufficiently large sample, for the hypothesis could not have pinned down the true value with full precision; in that case, one risks testing not the hypothesis but only whether one has taken a large enough sample.

Whereas the studies considered in the previous section involved mainly descriptive statistics, those to be considered now involve the determination of musical style, generally with a view to discrimination between possible composers or chronologies. The corresponding problem for literary texts is well-known and it was natural that the extension to music would be attempted. Statistical discriminant, factor, and cluster analyses have often been used for this purpose. Some early work was published by Schubert (1949), O. Winkler (1950), and Paisley (1964). Czekanowska (1986/1969) produced dendrograms on a somewhat ad hoc basis to provide a taxonomy of folksongs. Gabura (1970) made a thorough attempt to deal with the piano sonatas of Haydn, Mozart and Beethoven, claiming a fair but not high degree of success; one endorses his remarks (p. 224):

The analyst must draw on existing knowledge of music theory for guidance in choosing the pertinent musical parameters. . . . it is best to consider only statistics which can be interpreted musically.

Usher and Najock (1982), Crerar (1985) and Trowbridge (1985-6) dealt with earlier music, while Jacobson (1986) analysed over 40,000 bars of Schubert's music to help assign two undated works. Morando, (1979) used factor analysis on classical music, Steinbeck (1976) and Logrippo and Stepien (1986) cluster analysis on folksongs. Cope (1991), mentioned under Composition (section 2.2), also dealt with style determination.

The method of 'seriation' involves searching for that placement of given items on a straight line in which the distances between them best reflect their closeness according to measurements of their attributes. This method was used by Halperin (1992) in a novel attempt to decipher a fairly recently discovered musical notation from about 1500 BC, on the assumption that conjunct motion would have predominated in the melodies. Halperin (1994) later used the same method to study chronology among 13 troubadours.

Zipf (1949) studied the balance between the competing principles of unity and variety in several fields, taking an economic-statistical approach. His best-known result, 'Zipf's law', was derived originally for word usage in literary texts. Let N be the number of different words in a given text, and p(i) the relative frequency of the ith most commonly used word, for i = 1,...N; Zipf noticed that double-logarithmic graphs of p(i) against i were close to linear and so proposed that the following relation holds approximately over most of the range of i:

Thus a small number of words is used many times and a large number of words seldom, the mathematical relation being as prescribed. Although one might have thought this relationship would vary between authors, it turns out not to do so noticeably; it thus reflects a property of literary writing generally, not of individual styles. Zipf's first example is shown in Fig. 5.

Mandelbrot (1983: 344) suggested a second-order approximation with constants B and c, which however still reflects economy in general rather than distinguishing between authors:

This law might conceivably apply also to music. To look into that question requires first segmenting the music into contiguous sets of notes functioning as 'words'. Boroda (1988) undertook this difficult task, proposing an elaborate algorithm expressed in algebraic terms and taking into account duration and metre (see also Detlovs 1988). His level of success can be judged only on a musical basis and appears fair but not complete: in Fig. 6, for instance, the sixth and seventh notes belong together as appoggiatura and resolution and so should not have been separated; considering the relevant factors of pitch, harmony and accompaniment, a musically natural segmentation of the passage seems to be in groups of four notes throughout.

On the basis of the algorithmic segmentation the Zipf-Mandelbrot law was found to apply fairly well in many cases, although not in all, after visual comparison of the double-logarithmic graphs with the theoretical curve (Fig. 7). The properties of the statistical procedure for estimating the parameters might repay investigation.

Results in this area clearly have a bearing on the character of the relationship holding between musical language and spoken or written language; results in authorship studies (discussed above) and information theory (discussed below) have similar implications. Thus the stochastic theories of language, not discussed here, may be applicable to music.

Some contemporary composition proceeds by synthesis following statistical analysis of a given repertory. It is the analysis which interests us here.

The relative ease of numerically encoding melodies led to early attempts to apply information theory and Markov chains to them (the reader might wish to consult an elementary statistical text for an introduction to these topics). The analysis yields note-to-note transition probabilities for subsequent synthesis. The first such composition for computer was the Illiac Suite for String Quartet (1956; see Hiller and Isaacson 1959). Pinkerton (1956) dealt with tunes of nursery-rhyme simplicity; he measured redundancy for single notes and for note pairs and presented a composing algorithm based on note progressions considered separately for each metrical position. He mentioned without reference (p. 84) that Jean-Phillipe Rameau "made a statistical analysis of harmony similar in spirit to the approach we have used here", but my attempts to locate that analysis have so far been unsuccessful.

Cohen (1962) provided a critique of previous work and an extensive bibliography. A. and I. Yaglom (1967) also summarised previous work, but for them the calculation of information seemed more or less incidental. Knoppoff and Hutchinson (1981, 1983) discussed some formal aspects; Lisman (1986) wrote a short letter to the editor; Ames (1989) provided a survey and tutorial; Snyder (1990) recommended the use of high-order Markov chains, after it had become clear that low-order ones are inadequate. Much of music's important structure cannot be represented in these terms, however, so this approach seems unlikely to lead to real success and few workers today seem inclined to return to this field (but see Rhodes 1995-96).

Music can formally be described as a time series, whether single or multiple, but again it has so much important structure that traditional time series analysis is unlikely to be successfully applied to it. (Again the reader might wish to consult a statistical text for this relatively advanced topic.) For instance, music generally consists of phrases of possibly unequal length, which causes cyclical patterns operating over the course of a phrase to be re-phased many times in the course of a piece, and the common assumptions of statistical ergodicity and stationarity may be questionable (Cohen 1962: 155-157).

Nevertheless some attempts have been made, the most influential perhaps being that of Voss and Clarke (1978; see also Voss 1988) who claimed that much music is well modelled by a 1/f process, that is, one whose spectral density is approximately inversely proportional to frequency over a relevant range of frequencies; this was endorsed by Gardner (1978) as well as by Mandelbrot (1982: 375) who connected the claim with self-similar and fractal processes. These ideas were taken up by several composers (e.g. Bolognesi 1983; Dodge and Bahn 1986) and analysts (e.g. Hsü and Hsü 1990, 1991, who were criticized by Henderson-Sellers and Cooper 1993). Voss's claim was challenged by Nettheim (1992), who also pointed out the paradox that the resultant of a white-noise or 'random' series of durations and a white-noise series of pitches is by no means a white-noise melody.

Whereas Voss had analysed pitch and duration separately, Spyridis (1983), working independently of Voss, more appropriately analysed pitch and duration jointly. Pickover (1986) suggested using an ensemble of spectra with moving central point, represented in three dimensions, while Boon (1990) discussed complex dynamics generally.

The best-known example of chance applied to musical composition is the dice scheme attributed to Mozart (1787), in which two-bar units were carefully composed so as to give satisfactory musical results when played in random order; but aleatoric music in general will not be covered here, for the use of randomization techniques does not by itself provide strong statistical interest (Ames, 1987, 1990).

Also widely known is the work of Xenakis (1971, 1985). An application specially appealing to statisticians is his use of a large number of independent melodic lines to be played simultaneously, creating a resultant sound essentially different from that of each of its components; this phenomenon is analogous to the way in which the distribution resulting from the central limit theorem of statistics does not in general resemble the distributions of the components from whose addition it is formed. Xenakis (1971: 9) drew a different analogy:

. . . the song of cicadas in a summer field [is] made out of thousands of isolated sounds; this multitude of sounds, seen as a totality, is a new sonic event. This mass event is articulated and forms a plastic mold of time, which itself follows aleatory and stochastic laws.

Another influential contribution was made by Barlow (1980), who sought to parametrize many relevant features of his composition. His four central parameters were metric and harmonic cohesion, and rhythmic and melodic smoothness; these parameters constitute symmetrical pairs from the duration and the pitch domains. The entertainingly written paper introduces many appealing ideas of a statistical character, and has significance beyond the possibly slight musical value of the resulting composed piece.

A problem which Barlow encountered incidentally, which he called that of the 'persistent dispersion chain', is of some interest. His purpose was to specify points for note-attacks within a bar divided into n equal sections, n = 1, . . . 18; random allocation might by chance leave unwanted large sections empty of attacks, so he sought a method guaranteeing even dispersion. His example shows the idea (ibid.: 42-43): of the sequence

the first 2 numbers belong to different halves of the range 0-100, the first 3 to different thirds, and so on. He was able to find such sequences up to n=12 but not beyond. This problem is related to one considered by Steinhaus (1969: 49), who showed a surprising relation to the golden number mentioned earlier (section 2.1.3). I have found the following solution sequence for n=17 over the range 0-1000, but I believe there is none for higher n, an ironic limit considering that Barlow needed just n=18 for his musical purpose:

In Cope (1991) the statistical basis is less explicit than the computer methods but some novel features appear, including a simulated piece beginning in the style of Mozart and ending, after a gradual transition, in the style of Balinese Gamelan music (ibid.: 221-222); one's expectations in listening to this music should however not be too high.

If a musical performance follows the specifications in the printed score literally (without nuances of note duration or loudness), it will sound mechanical. An expressive performance, on the other hand, will modify these and other parameters in ways not spelled out in the score. The question exactly what makes a musical rendition suitably expressive was for a long time approachable only in subjective terms, but it has been reopened in the past decade with advances in the objective measurement of performance nuances. In particular, the exact length and loudness of each note played can be obtained from MIDI-compatible recording instruments such as the Yamaha Disklavier, or with more difficulty by examination of oscilloscope readings from pre-recorded performances (Repp 1990). Player-piano rolls also provide ready data (Husarik 1986). The performing standard of the subjects in such investigations is obviously crucial.

This area falls near the edge of the present paper's scope, and we refer to just one attractive idea which has emerged (Repp 1992): that rubato may tend to follow a parabolic course, differences of curvature reflecting different performers or musical contexts (Fig. 8). A remarkable anticipation of this discovery by 28 years may be found in a purely musical essay of Kirkpatrick (1984/1964: 68) who, from self-observation without any numerical data, referred to "the parabola that one describes in launching an idea".

Performance in a repertory lacking printed scores was studied by Will and Ellis (1994). They suggested that the intonation of some Australian aboriginal song is based not upon the logarithmic scale of frequencies previously thought to be universal, but upon a linear (although not equally spaced) scale. The verdict on this challenging suggestion has not been reached at the time of writing.

In many studies the data consists primarily not of the notes of musical scores but of responses to performed or recorded music, such as a judgement of the conveyed emotion. The responses may be of an individual or of a group, thus loosely belonging respectively to psychology or sociology. In such studies the statistical methods are unlikely to be specially shaped by the musical context, so only brief mention of these sub-fields will be made here.

The psychology of music is a large field making liberal use of statistical methods, generally conventional ones; a survey of that field would require a separate paper and we refer here to just a few representative studies. An early example is Watt (1924), dealing with the significance of the varying melodic intervals in two widely disparate song repertories (mentioned in section 2.1.3). Gundlach (1932, 1935) attempted to match various features of music to its emotional significance, first in American Indian music using descriptive statistics, and then in European art music using factor analysis. Rigg's series of papers (1937-48) pursued a similar aim using elementary statistical methods. Simonton (1977) used time series to study factors affecting creative productivity. Later (1980) he set out to find what makes a tune famous; using multivariate analysis restricted to the first six notes of the sampled themes he found (p. 206) that

(a) the fame of a musical theme is a positive linear function of melodic originality (rather than a curvilinear inverted-U function), and (b) melodic originality is a positive function of biographical stress and of historical time, and an inverted backwards-J function of age.

Connectionist methods, or neural networks, are a recent development taken up particularly by psychologists: Gjerdingen (1989, 1990), seeking pattern-perceiving procedures, is necessarily somewhat speculative, while the book by Todd and Loy (1991) provides an introduction to the field and examples. The question of what makes a musical rendition expressive rather than mechanical has also been taken up by psychologists (section 2.3).

The literature on the sociology and economics of music refers in larger part to popular than to art music; we will mention just a few items.

The social psychology of musical program-building was studied by Zipf (1946, 1949: 530-537), whose law concerning the economy of word usage was referred to in section 2.1.3. In the earlier of these two publications he looked into the question of the balance between familiarity, through the frequent re-programming of classical or staple pieces, and diversity, through the less frequent programming of modern or uncommon ones. His data consisted of the programmed items in 240 concerts in Boston over the years 1924-1934. It was to be expected that a small number of pieces would be performed fairly often and a large number of pieces only once or a few times. He found a specific form for this relationship (see Fig. 9): the number n of pieces of a given frequency f of performance closely followed a log-linear relationship with slope about -2:

Zipf studied similarly the frequency of performance of the works of the various composers and the distribution of the time interval between their re-use. Finally he explained the observed relationships by his theory of the opposing forces of uniformity and variety, a kind of economic theory (Zipf 1946: 31-35). In the later publication he showed, after summarising his earlier work, similar results concerning musical recordings and concerning the number of chamber music compositions written by various composers. As he noted, "the rectilinearity of the distributions is striking, since one does not normally expect to find an orderliness in an artistic phenomenon of this kind" (Zipf 1949: 532).

The question of the exhaustibility of the range of potential worthwhile and accessible musical styles was raised in a work of popular science by Gould (1996: 227-229). Gould's book deals throughout with statistical variation, but proposes no answer to the musical question.

Data is fairly readily available for further work in the sociology and economics of music, each country having its own sources of official or semi-official statistics. W. Winkler (1971) cited some German work of this kind in the 1930's, while Rubinstein et al. (1992) provided a comparison between Russia and the USA. An application to demography could investigate the often-remarked tendency of conductors and some other musicians to be long-lived, but I know of no systematic study of this question.

The earliest applications of statistics to musicology were made mostly in the field of ethnomusicology. Applications to Western art music date from about the 1930s, with psychologists among the early workers. Jeppeson (1946) is an early classic by a musicologist. Work on Zipf's law stands out for its novelty, but its significance is still to be established. In the 1950s and 1960s information theory was popularly applied to music; although it has not been specially successful efforts in this direction have continued sporadically. From the 1980s computer databases of music have been slowly developing, and the present difficulty of acquiring sufficient encoded data should be overcome within a few years. A number of approaches have been taken to their analysis, generally the counting of specified types of notes and patterns. Style discrimination, or the problem of authorship, currently appears to be one of the most fruitful areas—not just disputed authorship, but authorship characterization. Some claims have been made that music is a fractal process. Neural network modelling is one of the latest approaches at the time of writing.

Understandably, some musicologists have shown a lack of statistical expertise, while some statisticians have shown a lack of musical expertise; collaboration is often advisable. In any case, few applications of statistics in musicology have so far been fully convincing, as is not surprising at this relatively early stage. I believe that only when the structure of music itself is better understood and modelled by musicologists will statistical methods come into their own in this field. In the meantime, there is much scope for rewarding investigation.

This work was supported by an Australian Research Council Small Grant at the University of New South Wales.

Notes:
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