CH

AP

TE

R 6

Com

putational and Com

parative Musicology

Nicholas C

ook

Introd

uction

The m

iddle of the twentieth century saw

a strong reaction against the comparative

methods that played so large a part in the disciplines of the hum

anities and socialsciences in the first half of the century, and m

usicology was no exception. T

he term“com

parative musicology” w

as supplanted by “ethnomusicology,” reflecting a new

belief that cultural practices could only be understood in relation to the particularsocieties that gave rise to them

: it was sim

ply misleading to com

pare practices acrossdifferent societies, the ethnom

usicologists believed, and so the comparative m

usi-cologist w

as replaced by specialists in particular musical cultures. A

similar reaction

took place in theory and analysis: earlier style-analytical approaches (largely mod-

eled on turn-of-the-century art history) gave way to a new

emphasis on the particu-

lar structural patterns of individual musical w

orks. Perversely, this meant that the

possibility of computational approaches to the study of m

usic arose just as the ideaof com

paring large bodies of musical data—

the kind of work to w

hich computers

are ideally suited—

became intellectually unfashionable. A

s a result, computational

methods have up to now

played a more or less m

arginal role in the development of

the discipline.In this chapter I suggest that recent developm

ents in computational m

usicol-ogy present a significant opportunity for disciplinary renew

al: in the terms intro-

duced in chapter 1, there is potential for musicology to be pursued as a m

ore data-rich discipline than has generally been the case up to now

, and this in turn entailsa re-evaluation of the com

parative method. C

entral to any computational approach,

however, are the m

eans by which data are represented for analysis, and so I begin

with som

e examples of “objective” data representations before introducing the issue

of comparison. (T

he examples I discuss are graphic, but the sam

e points could havebeen m

ade in terms of num

erical representations.) This is follow

ed by an extendedcase study of an im

portant current software package for m

usicological research, theH

umdrum

Toolkit, and the chapter concludes with a brief consideration of the

prospects for computational m

ethods in musicology.

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Figure 6.1. Graphic analyses of Ligeti, Lux aeterna, section 2 (from Clendenning 1995: 948–49).

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Issues of R

epresen

tation

A picture, as everyone know

s, is worth a thousand w

ords. Each of the graphs in Fig-

ure 6.1 (from C

lendenning 1995: 248–

249) represents a different aspect of the same

section from Ligeti’s Lux aeterna: (a) charts pitch against tim

e, giving an imm

ediateim

pression of the music’s registral profile (b) show

s how m

any different pitches (notpitch classes) are present at any given point, and (c) highlights voice entries, w

itheach separate voice (four each of soprano, alto, tenor, bass) having a separate hori-zontal line w

hich is shaded whenever the voice enters, w

hile (d) shows essentially

the same inform

ation as (c), only in terms of the total num

ber of entries at any givenpoint. T

hese graphs are “objective” in the sense that, once you have agreed how a

graph is to be laid out, everyone should end up with the sam

e result; the information

is all there in the score, so that the analysis becomes, in effect, a m

atter of reformat-

ting. But the decisions about how to lay them

out are not necessarily self-evident.For exam

ple, in (a) to (c) the minim

um values on the tim

e axis are quavers, whereas

in (d) they are whole bars; other values w

ould have been possible, and the decisionto use these particular values represents a judgm

ent by the analyst that they will be

the most inform

ative in this context. There is also a degree of analytical judgm

ent inthe separate identification of the canons in (a) and (b).

That said, how

ever, it is obvious that there is scope for automation in graphs

like this—and not just in draw

ing them up (the graphs in Figure 6.1 w

ere gener-ated by a com

puter drawing package) but in the processing of the data. T

he graphsin Figure 6.2 (from

Brinkman and M

esiti 1991: 5, 6, 13, 17) were autom

atically gen-erated from

a machine representation of the score. 1

Again, each graph represents a

different way of extracting and view

ing the information in a score, this tim

e bars 1to 26 of the first m

ovement from

Bartók’s String Quartet no. 4: (a) corresponds to

Figure 6.1(a)(b)show

s each instrumental part separately w

ith the vertical lines pro-viding an im

pression of the linear movem

ent from one pitch to another, and

(c)charts the first appearance of each pitch, w

hile (d) represents the dynamic level

of each instrument (based on w

hether it is playing and on notated dynamic value)

and of the whole (in effect a sum

mation of the dynam

ic graphs of each instrument).

But what are w

e to make of such representations? W

hat do they enable us to see thatw

e can’t hear, and in any case, what is the point of seeing

music at all? It is a com

-m

onplace to describe Ligeti’s texture-oriented works as “visual,” and Brinkm

an andM

esiti (who in their article offer a case study of W

ebern’s Symphonie

Op. 21) em

-phasize the spatial sym

metry of W

ebern’s serial structures, comm

enting that “thegraphs allow

us to view all parts together in their spatial environm

ent. [This] is es-

pecially helpful since the pitch symm

etry that is essential to the musical structure of

this work is som

ewhat obscured by the notated score” (Brinkm

an and Mesiti 1991:

21). The question then is the extent to w

hich this problem of obscuring applies to

other repertories, in other words how

useful this kind of visualization is for music

in general.Yet this question is in som

e ways m

isguided: there is no such thing as a good orbad representation of m

usic per se, there are good and bad—

or at least better andw

orse—representations for particular purposes. To say that visual representations are

more appropriate for tw

entieth-century than for other repertories, or for answering

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certain kinds of questions about the music than others, is sim

ply to define their use-fulness, not to criticize them

. Nevertheless Brinkm

an and Mesiti produced som

e in-teresting results for m

usic that one wouldn’t norm

ally think of as particularly “visual,”including som

e strikingly different representations of music by Bach and M

ozart asw

ell as by Bartók, Schoenberg, Wolpe, and Berio. A

t the very least, such comparisons

serve to underline the variety of textures found in different composers or styles—

and texture is at the same tim

e one of the most im

portant aspects of music in term

sof how

we experience it, and one of the hardest to say anything useful about. It is not

hard to imagine how

Brinkman and M

esiti’s software could be used w

ithin a varietyof pedagogical contexts, enabling students to literally and instantaneously “see” dif-ferent pieces of m

usic in a variety of different ways—

and the ability to switch easily

and quickly from one representation of m

usic to another is a crucial element of prac-

tical musicianship. T

hat this has not happened, and that the potential of the approachhas not been fully realized, reflects the cost of translating academ

ic research projectsinto softw

are products for the real world, and illustrates an all-too-fam

iliar pattern inm

usicological software: a sustained burst of initial enthusiasm

is followed by running

out of money, resulting in softw

are that is sometim

es less than fully functional, oftenless than fully docum

ented, rarely properly supported, and usually soon obsolete.But m

y principal answer to the question “w

hat are we to m

ake of such repre-sentations?” is a different one. Brinkm

an and Mesiti (1991: 7) em

phasize the partic-ular usefulness of their graphs “as a device for com

paring different pieces”; Matt

Hughes, w

hose quantitative analysis of Schubert’s Op. 94 no. 1 w

as discussed inchapter 1, sim

ilarly claimed of his m

ethod that it was “a tool for organizing data so

that one may m

ore discernibly view tendencies and interassociations,” adding that

it was “best utilized w

hen viewing groups of com

positions or sections of a composi-

tion rather than a single work” (H

ughes 1977: 145, 150). And there is a general

point to be made here: the m

ore objective an analytical approach is, the more its m

u-sicological value is likely to be realized through m

aking comparisons betw

een dif-ferent pieces—

and sometim

es large numbers of them

. For example, w

hen you carryout a Schenkerian analysis, you are in a sense m

aking a comparison betw

een thepiece in question and the norm

s of comm

on-practice voice leading, as systematized

in the Schenkerian background and middleground. But the value of the analysis

consists primarily in the lengthy process of m

aking it, deciding which notes go w

ithw

hich, which are m

ore important than others, and so forth; the process is lengthy

because it involves a vast number of interpretive judgm

ents, requiring you to weigh

up different factors in relation to one another. At the end of it, you have a know

ledgeof the m

usic—you m

ight call it an intimacy

—that you did not have at the outset,

and there is a sense in which the final graph is significant m

ainly as a record of thislearning process. W

ith any kind of computational approach, by contrast, all of this

happens automatically, and in som

e cases almost instantaneously; the only output is

the graphic or numerical representation of the m

usic that results. And such repre-

sentations rarely tell you anything very useful by themselves: they m

ay well be vul-

nerable to the cheap but telling jibe that they either show you w

hat you hear any-how

(in which case they are redundant), or else they don’t (in w

hich case there areirrelevant). It is w

hen you compare representations of different pieces that useful in-

formation m

ay emerge.

Com

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(a)

(b)

(c)

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Com

putational and Com

parative Musicology

109

Figu

re 6.2.G

raphic analyses of Bartók, String Quartet N

o. 4, I, bars 1–

26 (fromBrinkm

an and Mesiti 1991: 5, 6, 13, 17).

(d)

The value of objective representations of m

usic, in short, lies principally in thepossibility of com

paring them and so identifying significant features, and of using

computational techniques to carry out such com

parisons speedily and accurately. At

this point, however, the issue of reduction (already broached in chapter 1, this vol-

ume) has to be confronted. T

his can conveniently be explored in relation to theanalysis of folksongs, since a large of proportion of early w

ork in computational

analysis was based on such m

aterial: there were substantial existing collections of

folksongs to work on, and the m

usic was m

onophonic and scalar, hence could becoded very com

pactly—

a vital consideration in the days when com

puter mem

oryw

as a scarce and expensive comm

odity. But of course, if you code folksongs as aseries of pitches, or scale degrees, then you are leaving out all the variables of into-nation, ornam

entation, timbre, and other aspects of singing style, not to m

ention the

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original contexts of use and cultural connotations of the songs. And that obviously

limits w

hat you can conclude from com

paring them.

At this point a concrete exam

ple may be helpful, and Figure 6.3 is taken from

the Essen M

usical Data Package, a series of databases of popular and traditional

vocal repertories together with tools for encoding and analysis. 2

Coordinated until

his death in 1994 by Helm

ut Schaffrath, the Essen databases consisted by the fol-

lowing year of som

e 10,000 songs, mainly E

uropean (and largely Germ

an), but with

some representation of other traditions, particularly C

hinese; they are still beingadded to, though developm

ent has been transferred from E

ssen to Warsaw

(Self-ridge-Field 1995). Figure 6.3 show

s “Deutschland über alles” in the custom

codeused by the package, EsA

C(the E

ssen Associative C

ode), and several of the datafields are self-explanatory: BO

EH

ME

is the name of the database in w

hich this songis located, C

UT

contains the title, RE

G is the region from

which it com

es, and FCT

is the functional category of the song, while K

EY contains not only its key (F) but

also the numbering of the song in the database (B0001), the value of the sm

allestnote value (16, a sixteenth or sem

i-quaver), and the time signature (4

/4). The tune

itself is contained in the field ME

L, using a simple code based on the scale degree

(with # for sharp and b for flat degrees), �

and �to indicate shifts to a higher or

lower register, respectively, and a rhythm

ic notation based on the shortest value(sem

i-quaver), with a dot prolonging the value by 50 percent, a single underline

character (_) by 100 percent, and a double underline character (_ _

) by 200 percent.T

hat should suffice for the tune to be read; the only other information contained in

the ME

L field is //(for the end of the m

elody), the use of spaces to indicate bar lines(the m

elody has been notated across the bar lines), and the splitting up of the mel-

ody into separate lines, each of which represents a phrase (this represents a judg-

ment on the part of the transcriber, and as such is a rather unusual feature of the

EsAC

code).W

hat can be done using this information? T

he distributed version of the Essen

database comes w

ith a simple analytical utility (A

NA

), which takes a com

plete data-

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base as its input, and generates as its output an annotated version of the file in which

up to 12 different kinds of analytical information are added for each song. T

hese in-clude straightforw

ard statistical information such as the distribution of ascending

and descending intervals (how m

any ascending minor seconds?), the distribution of

durations (how m

any crotchets, quavers, semi-quavers?), and distribution of scale

degrees (how often does the fifth scale degree appear in the low

er register?); thereare also som

e derived measures such as the percentage of ascending steps and leaps.

They also include som

e less obvious information, including the pattern of phrase

repetitions in pitch and duration, a coding of the contour of each phrase, them

elodic spine (based on the notes at metrically stressed beats), and the final notes

of each phrase (this is where the coding of phrases com

es in). 3A

limitation of the

AN

A softw

are is that it provides data for each song separately, whereas the statistical

information w

ould be more useful if it w

ere provided across the database as a whole,

but there is a reason for this: the package was intended for use w

ith a comm

ercialrelational database package, w

hich would facilitate this kind of analysis. N

eedless tosay, the original package is no longer obtainable. 4

But what can be done

using this information? Schaffrath him

self published anum

ber of studies based on his datasets, in which he used this kind of analytical in-

formation to support broad stylistic generalizations, for exam

ple as between differ-

ent traditions: “Germ

an folksongs,” he concluded from one such study (1992: 107),

“generally skip more often up than dow

n; in the Chinese pool the opposite ap-

plies.... Germ

an songs use the interval of a fourth 51 percent more often than C

hi-nese songs do. T

his might be explained by the preference for pentatonic scales.” But

there is also a different way in w

hich such information can be used, w

hich is for pur-poses of searching, identification, and classification. A

database of 10,000 folksongsis not unlike a bibliographic database, and to locate individual item

s or groups of re-lated item

s you search for particular features, combining them

so as to narrow the

search down to a m

anageable number of hits. In som

e contexts all you need to searcha bibliographic database is the author’s nam

e; in other cases you need to combine an

author name, tw

o title keywords, and year to locate the article you need. In the sam

ew

ay, you might try to locate a particular song by searching for its first few

notes orincipit; this is the w

ay that traditional dictionaries of themes operate (for instance,

Barlow and M

orgenstern 1983), and it works w

ell with “art” m

usic, where there are

fixed scores. But it is not such a good way to search for folksongs, w

hich typicallyexist in a large num

ber of variants, often involving variation or the interpolation ofnotes. T

hat is why A

NA

includes a routine to derive the final notes of each phrase: asSchaffrath (1997: 349) explains, “although variants m

ay contain different numbers

of notes or melodic contours, they tend to retain underlying harm

onic structures.”O

r you could combine such a search w

ith a particular melodic spine and

/or patternof phrase repetition. A

nd whereas I have described the use of such features in term

sof sim

ply locating items, the sam

e processes can be used to group together songs orrepertories in w

ays that might reflect, for exam

ple, historical development, linguis-

tic associations, or conditions of use. In such ways the autom

ated analysis of musi-

cal style can become a m

eans of generating or verifying musicological hypotheses.

Or at least that is the aspiration. In practice, EsA

Cand A

NA

suffer from obvious

limitations, ranging from

the pragmatic (the output of A

NA

is only marginally m

ore

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user-friendly than machine code) to those of principle: EsA

Ccan cope w

ith only anarrow

range of music (m

onophonic, restricted to three octaves, and so on), and isw

ell adapted for only a narrow range of applications. A

NA

extracts a limited num

berof features, each of w

hich represents a radical reduction of the music and is intended

for a particular purpose (such as statistical generalization or pattern matching);

moreover, the usefulness of these reductions depends on a num

ber of quite specificassum

ptions, such as that the final-note patterns of phrases vary less than incipits.A

ll this makes sense in term

s of what the E

ssen package is designed for; it is, to adoptoutdated softw

are jargon, a “turnkey” solution to folksong analysis, or at least to acertain sort of folksong analysis. A

nd it would be possible to im

agine trying to de-velop the code, and the analytical softw

are, in such a way as to be m

ore flexible—to cope w

ith music of any desired degree of com

plexity and variety, and to extractfrom

it every feature that could be of any possible interest under any possible cir-cum

stances. That w

ould be like the approach of business software developers in the

1980s, who aim

ed at prepackaged, turnkey solutions in which every eventuality had

been foreseen and the appropriate feature bolted on somew

here, if only you couldfind it.

In musicology, as perhaps also in the office, this represents a basic m

isappre-hension

—in term

s not only of the unlimited variety of m

usical materials, but also

of the variety of purposes for which people w

ill want to represent them

; as Huron

(1992: 11) puts it, “The types of goals to w

hich music representations m

ay serve arelegion and unpredictable.” C

omplexity is not the solution. T

here are, after all, much

more com

plicated codes than EsAC

which fulfill certain purposes extrem

ely well

while being very poorly adapted for others. M

IDI code has transform

ed entire sec-tors of the m

usic industry, but because it tries to understand all music on the m

odelof keyboard m

usic its handling of microtonal inflection is inelegant, to say the least;

it also doesn’t distinguish enharmonic equivalents, and has to be drastically recon-

figured if useful analytical information is to be extracted from

it—even som

ethingas sim

ple as locating a C m

ajor triad. 5A

gain, DA

RM

S6

is an extremely pow

erful codedesigned for purposes of printing and publication (it is the basis of all A

-R E

ditionscores), and accordingly it understands m

usic as a kind of graphics; it doesn’t thinkof an e

1but of a notehead on the bottom

line of a stave that has an F clef on it. As a

result, it too needs drastic reconfiguration if analytically useful information is to be

extracted from it (Brinkm

an 1986). And though the problem

s of multiple codes are

alleviated by the existence of interchange utilities (e.g., to turn DA

RM

S into MID

I),there are lim

itations to what is possible: since M

IDI does not distinguish betw

een C �and D �, the inform

ation is simply not there to translate into D

AR

MS, w

hich doesm

ake the distinction.A

s with business softw

are, the solution to this problem of m

ultiple requirements

is not the creation of ever more com

plex and unwieldy integrated solutions, but a

modular approach involving an unlim

ited number of individual softw

are tools de-signed to serve individual purposes; that w

ay, when you w

ant to do something new

,you sim

ply design a new tool to do it. M

odular approaches like this, however, need

some kind of fram

ework w

ithin which to w

ork—

in essence, a set of rules for therepresentation of data in m

achine-readable files, and for the transfer of information

between different softw

are tools. Hum

drum is an exam

ple of just such a framew

ork.

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A C

ase Study: T

he H

um

drum

Toolkit

Developed by D

avid Huron in the early 1990s, H

umdrum

7consists of tw

o distinctelem

ents: in the first place a syntax—

the set of rules I referred to—

and in the sec-ond, a “toolkit” consisting at present of som

ething over 70 separate software routines

for manipulating the data in different w

ays for different purposes. It runs in a UN

IXenvironm

ent, 8and its m

odular approach is much m

ore readily understood by thosew

ho have experience of UN

IX than those w

ho do not. Like Hum

drum, U

NIX

can bethought of as a set of rules plus a set of tools: individual U

NIX

comm

ands (which

have names like grep, sort, and cat) do sim

ple things like searching files for a partic-ular string and extracting, deleting, or altering it, or rearranging data w

ithin or be-tw

een files. The pow

er and flexibility of the system com

es not from these individual

tools but from the possibility of “pipelining” them

so that the output of one tool be-com

es the input of the next, forming chains of com

mands of unlim

ited length. The

Hum

drum toolkit is sim

ply a set of additional UN

IX tools designed specifically for

manipulating m

usic-related data. It is this open and modular philosophy, w

hich al-w

ays allows for an additional tool to be added w

hen required, that explains why

—as H

uron (1997: 375) puts it—“H

umdrum

is especially helpful in music research

environments w

here new and unforeseen goals arise.” U

nfortunately, as we shall see,

it also means that the user has to have a detailed understanding of the w

ay the dataare encoded and the w

ays in which the tools m

anipulate it. Hum

drum, in short, has

a steep learning curve.A

ll this may sound very abstract, so Figure 6.4 show

s the first two phrases of

“Deutschland über alles” in kern

code, the normal H

umdrum

format for represent-

ing notated music (only tw

o phrases because when printed out—

though not interm

s of internal storage—kern

is much less com

pact than EsAC

). 9T

his is a directtranslation into kern

of Figure 6.3, so the same inform

ation may be found in the data

fields preceded by !!! (the exclamation m

arks identify them as com

ments); only the

information concerning Schaffrath, the copyright statem

ent, and the !!!AR

L line(w

hich gives latitudinal and longitudinal coordinates) are new. A

s for the other lines,**kern m

eans that what follow

s is in kern(the double asterisk m

eans that this labeldefines the data that follow

s), and the lines preceded by a single asterisk define theinstrum

ent category and instrument (in each case “vox” or voice), m

eter (4/4), key

signature (B �, the “flat” being designated by “�”—

the sign for sharp is “�”), and

key (F). 10T

he tune itself begins on the following line, preceded by a curly bracket

(this indicates the phrases, while the lines beginning w

ith “�” are bar lines). “4.f”

means that the first note is the F above m

iddle C (if it w

ere an octave lower it w

ouldbe F, if an octave higher ff, if an octave higher than thatfff, and so on); the “4” m

eansthat it is a quarter note (crotchet), and the “ .

”, unsurprisingly, that it is dotted. Itshould now

be possible to read the melody quite straightforw

ardly. (The apparently

simplistic reciprocal notation for durations is unexpectedly flexible, coping w

ithm

ost things short of Ferneyhough; triplet quavers, for instance, work out as 6, quin-

tuplet semi-quavers as 20). T

hat explains everything in Figure 6.4, except the “ * ”,w

hich marks the end of the record. It should also explain everything in Figure 6.5,

which show

s the same phrases (w

ith the comm

ents omitted), as set in G

major by

Haydn in the second m

ovement of his String Q

uartet Op. 76 no. 3: instead of the

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single column (or spine) of Figure 6.4, there are now

four, one for each line of them

usic. It is helpful to think of this as being laid out like staff notation, only rotatedby 90 degrees.

Kern

is not as easy to read as EsAC

(which, apparently, w

as designed to be easyto sight read). A

nd it would seem

to be better adapted for linear textures than, saykeyboard m

usic—though in fact there is no lim

itation in this respect, partly becauseyou can put m

ore than one note on a single line within any one spine, and also be-

cause you can collapse spines into one another or create new ones at any point. It is

also more flexible than it looks. In the first place, you don’t have to specify every-

thing in Figures 6.4 or 6.5; there is no need to encode phrases, or bar lines, or du-rations, or pitches (in fact the only

mandatory lines are “**kern” and “*

�”). In the

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second place, there is a large number of further codes and interpretations built into

kern, allowing you for instance to include sym

bols for articulation or ornamentation,

clefs, stem direction, or the num

ber of staff lines, if any (and whether they should

be colored or dotted). But more than that, **kern spines can be m

ixed with others,

including, for example, such predefined representations as **text (for lyrics) or

**IPA (lyrics, but now

notated using the International Phonetic Alphabet), or new

representations defined by the user: you might, for exam

ple, create spines labeled**bow

ing, to show how

a given pattern is (or might be) bow

ed, or **timing, to en-

code the rhythmic nuances in a particular perform

ance, or **heartbeat, to record lis-teners’ physiological responses to hearing the m

usic. Then again, you m

ight not use**kern at all, but an alternative representation for pitch; predefined representationsinclude **pitch (using the International Standards O

rganization format), **deg

(scale degrees, as in EsAC

), **solfg (French fixed-do solfège), **freq (frequency

Com

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expressed in cycles per second), or the self-explanatory **midi; any of these repre-

sentations can be automatically translated to any other, provided of course that the

relevant information is there to be translated (recall the problem

with M

IDI and en-

harmonic equivalents). T

here is also a particularly well-developed **fret represen-

tation, which turns pitch notation into tablature notation based on any specified

number of strings and tuning, or of course you m

ight always invent a new

repre-sentation for som

e particular repertory or purpose.A

nd what does all this enable you to do? For a start, you can replicate the func-

tions of AN

A: to extract the m

elodic spine of Figure 6.4, for instance, you locate thedow

nbeat of each bar (search for lines beginning “�” and then skip to the next line),

skip over the duration figures, and extract the pitch symbol(s) to a new

file. Or you

could just as easily search an entire “BOE

HM

E” database for m

elodic motions from

^4 to ^5 and establish how frequently they occur by com

parison with the reverse

motion (you w

ould translate the database into **deg records in order to do this, andsearch for adjacent lines containing 4 and 5, skipping over irrelevant sym

bols likedurations, and finally counting the num

ber found). Or you could search all of Bach’s

chorales (the Riem

enschneider collection is available in kerncode) in order to es-

tablish how often the m

elodic leading-notes are approached from beneath, as against

from above: to do this, you extract the spines containing the m

elodies, turn theminto **deg records, and count the occurrences of “^7” as against “v7.” 11 (T

his par-ticular task is easier than it sounds, because **deg autom

atically codes the directionin w

hich a given scale degree was approached, though not the size of the interval in-

volved). Then again, shifting to a different level of com

plexity, you could analyze therelationship betw

een a particular succession of vowels in song texts and the m

elodicor rhythm

ic contexts within w

hich they occur. You could establish how far particu-

lar harmonic form

ations are correlated with specific m

etrical locations (and analyzethe results by com

poser, period, or geographical origin). Or you could classify dif-

ferent melodic, harm

onic, or structural contexts in Chopin’s com

plete mazurkas,

and use this as the basis for analyzing timing inform

ation extracted from a large

number of recordings.But the best w

ay to see what H

umdrum

is capable of is to examine published

studies that have used it, of which the m

ajority (though by no means all) are by

Huron and his co-w

orkers; I shall briefly describe five. An article on the m

elodic archin folksongs (H

uron 1996) is based on the Essen collection and provides a direct

comparison w

ith Schaffrath’s own w

ork. Its purpose is to test systematically the fre-

quent, but inadequately supported, claim that “arch” contours are prevalent through-

out Western folksongs, and H

uron points out the dangers of “unintentional bias”(1996: 3) w

hen such generalizations are supported by a small num

ber of possiblyhandpicked exam

ples: they can be firmly established only through the analysis of

data sets large enough to allow the draw

ing of statistically significant conclusions.H

e puts forward tw

o basic ways in w

hich folksongs might be said to exhibit arch-

shaped contours—w

ithin each phrase, and overall—and system

atically analyzesthe E

ssen collection for each. For the first test, individual phrases throughout thecollection w

ere sorted according to the number of notes in them

, and the averagecontour for each phrase-length com

puted; for the second, the average pitch of eachphrase of each song w

as computed, and the resultant contour classified. T

he overall

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conclusion was that there is a strong tendency tow

ard arch-shaped contours on bothm

easures (though a third approach, based on the nesting of arches within one an-

other, was not supported).

A project like this m

ight be described as regulative: it takes an existing musico-

logical claim and tests it rigorously against a large body of relevant data, adopting cri-

teria of significance and certainty that may be unfam

iliar in musicology but are nor-

mal in data-rich disciplines. Tw

o further studies illustrate the range of musicological

claims that can be tested in this m

anner. A study carried out in collaboration w

ithPaul von H

ippel (Hippel and H

uron 2000) focused on the “gap-fill” model of m

elody,according to w

hich listeners expect melodic leaps to be follow

ed by a change of di-rection; this is an im

portant element of N

armour’s “im

plication-realization” theory,w

hich seeks to explain how listeners experience m

usic on the basis of whether or not

implications set up by the m

usic are realized in what follow

s. What is at issue is not

whether or not m

elodic leaps are usually followed by a change of direction, w

hich isundoubtedly the case; it is w

hether or not this happens (as Krum

hansl has concludedon the basis of experim

ental studies) 12because of listener expectations. T

he alterna-tive is that it is a trivial consequence of the lim

ited registral range of vocal music, a

possibility which von H

ippel and Huron (2000: 63) explain by com

parison with

a simplified m

elody that is confined to a range of three adjacent pitches—for exam

ple, A, B, and C

. In such a melody, the only skip available is betw

eenA

and C. U

pward, this skip m

ust land at the top of the range; downw

ard, itm

ust land at the bottom. A

fter a skip, therefore, there is no way for a m

elodyto continue m

oving in the same direction. O

n the contrary, two of the three

available pitches can be reached only by a reversal. Although m

ost melodies

have a wider range, they are subject to the sam

e basic argument.

If this second possibility is the case, then gap-fill patterns will be no m

ore comm

onin real folksongs than in random

ly generated melodies w

ith otherwise com

parablecharacteristics—

and, to cut a long story short, this is exactly what von H

ippel andH

uron found. If their argument (w

hich I have grossly simplified) is accepted, then

it provides a striking instance of how studies based on large data sets can raise ques-

tions about conclusions derived even from experim

ents as meticulously controlled

as Krum

hansl’s. 13

A third study (H

uron 2001) makes a sim

ilar kind of point in relation to “art”m

usic. It reassesses Allen Forte’s (1983) set-theoretical analysis of the first m

ove-m

ent of Brahms’s String Q

uartet Op. 51, no. 1

—an application to high R

omantic

music of an approach originally developed for tw

entieth-century music, bringing

with it an appearance of objectivity and rigor in stark contrast to conventional m

o-tivic descriptions of such m

usic. (In other words, Forte’s article is a late expression

of the postwar culture of objectivity described in chapter 1, this volum

e.) Whereas

a “motive” is sim

ply a characteristic melodic figure that recurs prom

inently through-out a piece, Forte’s “alpha” ic (interval class) pattern can be defined m

uch more pre-

cisely: it is a particular intervallic configuration that may occur in any of the stan-

dard transforms of set theory (prim

e, inversion, retrograde, retrograde inversion), inany rhythm

ic configuration, and at any metrical point. So H

uron proceeds to searchsystem

atically for Forte’s “alpha” pattern, not only in the first movem

ent of Op. 51,

Com

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No.1 but also in the first m

ovements of Brahm

s’s other quartets—a com

parablerepertory in w

hich the “alpha” pattern, if it is really characteristic of Op. 51 no. 1,

should be significantly less comm

on. First, he demonstrates that, as defined by

Forte, the “alpha” pattern is comm

on in Op. 51 no. 1, but no m

ore so than in theother quartets. H

owever, if you restrict the search to instances of the pattern that

begin on metrically strong beats and com

e at the beginning of slurs, then it is more

prevalent in Op. 51 no. 1 than in the others. T

his is particularly the case if you lookfor the pattern only in its prim

e form, and even m

ore so if you consider only thosecases w

here the pattern coincides with a long-short-long rhythm

ic pattern. So theprevalence of Forte’s “alpha” pattern is confirm

ed—

but only under precisely thoseconditions that are captured by the conventional idea of a m

otive! The conclusion

must be that the appearance of rigor in Forte’s analysis as against traditional analyti-

cal descriptions is just that, an appearance.Im

portant and indeed salutary as this regulative function may be, it w

ould bew

rong to give the impression that com

putational analysis is good only for testing thevalidity of existing theoretical or analytical claim

s. It can also be used to undertakew

ork that could hardly be achieved in any other way. A

spectacular example is the

correlation of musical features w

ith geography, a project once again based on theE

ssen database (Aarden and H

uron 2001). Recall that the E

ssen records include in-form

ation concerning the geographical origins of the songs contained in them—

sometim

es down to the level of an individual tow

n or village, and that the kernver-

sion of “Deutschland über alles” added longitudinal and latitudinal coordinates in

the !!!AR

L field (a new field defined specifically for A

arden and Huron’s project).

Standard Hum

drum com

mands can be used to extract records from

the databaseand output their coordinates; these are then used as input to a m

apping program. By

way of exam

ple, Figure 6.6 shows the distribution of m

ajor- and minor-m

ode songsw

ithin the Essen database; these m

aps were generated using the G

EO

-Music site

(http://ww

w.m

usic-cog.ohio-state.edu/cgi-bin

/Mapping/m

ap.pl), established as partof the project. It is hard to know

what conclusions m

ight be drawn from

this com-

parison (other than that the major m

ode is considerably more w

idespread than them

inor); the southward skew

ing of the major-m

ode data as against the minor sim

plyreflects the three occurrences in Italy

—hardly a basis for robust generalization

—w

hile the concentration in Germ

any and central Europe evident in both cases re-

flects the bias of the Essen database. A

fully fledged musicological research project

would no doubt involve dealing w

ith more features at a greater level of detail. N

ever-theless this exam

ple does indicate the potential for computational m

ethods to drawtogether quite diverse kinds of inform

ation; it also illustrates the value of graphicpresentation of the com

plex data generated by work in com

parative musicology.

A final study (H

uron and Berec, forthcoming) is perhaps even m

ore suggestivein term

s of possible musicological applications. Like H

ofstetter’s attempt (m

entionedin chapter 1) to turn a claim

about the “spirit of nationalism” into an em

piricallytestable proposition, H

uron’s and Berec’s starting point is a woolly, com

mon-language

concept: idiomaticism

. How

might you set about defining w

hat is idiomatic on, say,

the trumpet in such a w

ay that a computer could m

ake evaluations of just how idio-

matic a particular piece of m

usic is? In the first place, Huron and Berec say, idiom

at-icism

is not the same as difficulty: a piece of trum

pet music can be difficult but idio-

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matic, or easy and unidiom

atic. It is rather a matter of achieving the desired m

usicaleffect in the easiest possible m

anner. For example, a piece w

ill be idiomatic if it is sig-

nificantly easier to play as written than w

hen transposed up or down by say a sem

i-tone; this is “transposition idiom

aticism.” So H

uron and Berec evaluated the diffi-culty of a num

ber of pieces when played in all transpositions from

an octave belowthe original to an octave above, and in several cases the original proved to be signif-icantly easier to play than m

ost or all of the transposed versions. (They also ran a

parallel test based on how m

uch more difficult the m

usic was w

hen played at a dif-ferent tem

po from the notated one, w

ith similar results.) But there w

as also a furtherfinding: the pieces had been selected so as to include exam

ples by both trumpet vir-

tuosi and nontrumpeters, and the conclusion in respect of both transposition and

tempo idiom

aticism w

as the same—

that expert trumpeters com

pose more idiom

at-ically than com

posers who cannot play the instrum

ent.T

hat conclusion might be considered predictable, not to say obvious (just as in

the case of Hofstetter’s discovery that there are differences betw

een national styles).A

nd it would have been of lim

ited interest if Huron and Berec had sim

ply askedtrum

peters to play the pieces at different transpositions and at different tempos, and

to say how difficult they w

ere, an approach that would not have required H

umdrum

,of course. But instead

—and this is the real point of their study

—they attem

ptedsom

ething much m

ore challenging: what they refer to as “the design and im

ple-m

entation of a computer m

odel of a trumpet perform

er.” Their m

odel takes a kernfile as its input, and evaluates the difficulty of perform

ance along a variety of differ-ent dim

ensions: in terms of the valve transitions betw

een successive notes; in terms

of register, particularly in the case of sustained notes; in terms of dynam

ics; and interm

s of tonguing at different speeds. (The inform

ation on which they based the

model included perform

er assessments in the case of valve transitions, tests of per-

formance in the case of tonguing, and physiological data concerning lung capacity

and diaphragm support in the case of sustained notes.) T

he model w

as tested usinga series of trum

pet studies set for different grades by the Royal C

onservatory ofM

usic of Toronto; there was a generally fairly high correlation betw

een its estimates

of their difficulty and the Royal C

onservatory’s. But of course the main test w

as theevaluation of idiom

aticism, and the fact that the m

odel come up w

ith what every-

body knows confirm

s the extent to which it actually w

orks. That is the real conclu-

sion to be drawn from

the study.D

o musicologists really need

a computer m

odel of a trumpet perform

er? One

answer to this is that the approach could be readily generalized to other instrum

ents:to the recorder (w

here finger transitions would be of particular im

portance, owing

to the complexity of cross-fingerings), to the piano (the m

odel might propose the

best fingering, which could be tested against those specified by editors or used by

performers), or to the guitar (in effect a m

odeling of the dance of the fingers on thefretboard). But the m

ore substantial answer lies behind this. O

ne of the chronicproblem

s with analyzing m

usic is the tendency toward abstraction: the m

ore so-phisticated our analytical m

odels, the more divorced the object of analysis seem

s tobecom

e from the experience of m

usic, and especially from the sense of physical en-

gagement in w

hich much if not all m

usic has its source. It is conventional to deplorethis, but to do nothing about it. In this context a com

puter model of the electric gui-

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tarist’s left hand could become, perhaps paradoxically, a m

eans of thinking the bodyback into m

usical analysis: you could, to give just one example, chart the interaction

of the dancing fingers with the ear—

of the imperatives of the fretboard w

ith thoseof “purely m

usical” implication and realization

—in rock guitar im

provisation. Inthis w

ay computational m

usicology holds out the promise not only of providing

more robust answ

ers to questions that have already been asked, but also of making

it possible to ask new questions.

Con

clusion

: Brave N

ew W

orld?

Promises, prom

ises... and it m

ust be admitted that com

putational software has a

hardly better record of delivering on its promises than dot com

companies. Let’s stick

with H

umdrum

as the most likely current candidate for a pow

erful, general-purposecom

putational aid for musicology, and explore som

e of the practicalities of doingm

usicology with it.

Q. Is it easy to get hold of?

A. Yes, at the tim

e of writing you can dow

nload it for free, or you can buy itfor the cost of the m

aterials from the C

enter for Com

puter Assisted

Research in the H

umanities at Stanford. 14

Q. W

ill it actually run on my com

puter? Don’t I have to be using a U

NIX

machine?

A.You can run it on a PC

or a Mac, but you w

ill need to install a UN

IXtoolkit. You can probably get that for free, too. 15

Q.Butw

illI be able to find the music I w

ant to work on in kern

code?A

. That depends w

hat you want to w

ork on. The E

ssen package is available in kern, and so is som

e of the Densm

ore collection of Native A

merican

music, so if you are interested in folk m

usic you can start work right

away. A

s for Western “art” m

usic, there is quite a lot out there: not onlyBach’s chorales but also m

ost of his cantatas, as well as the W

ell-Tempered

Clavier; substantial am

ounts of Corelli, V

ivaldi, Handel, and Telem

ann;cham

ber and orchestral music by H

aydn and Mozart; and m

ost of theBeethoven sym

phonies. 16T

he hope, of course, is that as more m

usicolo-gists use H

umdrum

, so more m

usic will be encoded, encouraging m

orem

usicologists to use Hum

drum—

and in this way you get a virtuous

circle. (The sam

e applies to the development of the tools them

selves.)Q

.And if I can’t find w

hat I want?

A. If it exists in som

e other code you may be able to translate it into kern.

Translation routines exist for the MuseD

atacode used by the C

enter forC

omputer A

ssisted Research in the H

umanities (C

CA

RH

), the Plaine andE

asie Code used for R

ISM (R

épertoire international des sources musicales),

and MU

STR

AN

as well as the code used by Leland Sm

ith’s music notation

program SC

OR

E, and the “E

nigma” code used by Finale. N

ot all of thesetranslators are readily available or unproblem

atic, but if you still can’tfind w

hat you want you can of course encode your ow

n. Rather than

doing it by typing in the code, you can use an encoding routine that

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enables you to enter the music on a M

IDI keyboard

—rather like step-

time entry in a sequencer package. T

here are also utilities for playing kern

code (via MID

I), for displaying music notation on screen, or for out-

putting it as a PostScript file (though not all of these are available if youare running H

umdrum

on a PC or a M

ac). You can turn kernfiles into

Finaleones for printing, too.

Unfortunately, none of this addresses the real difficulties of integrating H

um-

drum into everyday m

usicological life. The fact is that not everybody is happy w

itha U

NIX

comm

and-line environment, or finds it easy to rem

ember the different com

-m

ands with their m

ultiple (and often complex) options; people w

ho use UN

IX (or

Hum

drum) on a daily basis can operate it m

uch more quickly than a graphic envi-

ronment (like W

indows, w

ith its drop-down m

enus and mouse control), but if

Hum

drum is to becom

e part of everyday musicological life then w

hat matters is its

accessibility to the occasionaluser. Michael Taylor (1996) has developed a graphical

user interface (GU

I) for Hum

drum, though it is not publicly distributed at the tim

eof w

riting, 17and this m

akes life considerably easier for the occasional user: there aredrop-dow

n menus listing the various H

umdrum

comm

ands and dialog boxes where

you set the associated options, and you can even select musical elem

ents using ordi-nary language (“any bar line,” “at least one flat,” or “C

major chord,” for instance). 18

For nonspecialist users such an interface—if generally available and supported

—w

ould surely represent a substantial step in the right direction: it not only looks fa-m

iliar but also constantly reminds you of the com

mands and options that are avail-

able, without significantly com

promising H

umdrum

’s functionality and flexibility.But such an approach can only go so far. T

his is because, to use Hum

drum at

all, you need quite a detailed knowledge of its representations (kern

and the rest); agreat deal of H

umdrum

usage has to do with things like extracting the spine you are

interested in, merging spines, or stripping out inform

ation that is not wanted for any

particular operation—

and even ordinary-language definitions cannot protect youfrom

the need to understand what is going on in the code. For this reason A

ndreasK

ornstädt (1996) advocates a different approach: instead of “a Hum

drum‘com

mand

center’ with lots of buttons and gadgets w

ith names like yank

and MID

I|smf

,” ashe puts it (1996: 119), he has developed an analytical environm

ent into which dif-

ferent GU

I modules can be slotted for different applications. E

ach module can be

thought of as a “browser” that let users view

just those musical elem

ents that theyare interested in, and no others; for instance, K

ornstädt has created such a systemfor leitm

otivic analysis, in which users can define leitm

otifs and locate their occur-rence in a score on the basis of on-screen notation and intuitive com

mands. T

his,again, is not publicly available, but another of K

ornstädt’s customized applications

of Hum

drum is: an on-line them

atic dictionary, called Them

efinder. 19You type in

some pitches from

the tune you want to find (or alternatively you can type in the

scale degrees, or the intervals between the notes, or even just its contour), and the

software finds all the m

atches in its database and displays them on screen in m

usi-cal notation. T

here are no Hum

drum com

mands; you don’t even need to know

it isH

umdrum

that is doing the work

—and w

hat makes this possible is that T

heme-

finder is designed to do one thing and one thing only. It is hard at the present time

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to tell how far K

ornstädt’s approach—

the development of specialized system

s fordifferent analytical purposes—

will prove to be com

patible with the flexibility and

unpredictability of real-world m

usicological research, or whether it represents a par-

ticularly sophisticated way of falling into the sam

e traps as 1980s office software.

There is also the question of how

the substantial costs of developing and updating acom

prehensive system of this kind are to be found.

Maybe, in the com

ing years, the combination of user-friendly interfaces for

Hum

drum and a developing body of m

usicological work using it w

ill provide thenecessary m

omentum

for it to become m

ore generally used, so that in time it w

illbecom

e just one of those skills that musicologists have to acquire (like using nota-

tion software, for exam

ple). Or m

aybe that is just not going to happen; in that case,com

putational software like H

umdrum

might still becom

e part of the wider m

usi-cological scene, but as a specialist professional service, rather than a skill you acquirefor yourself (rather like note processing w

as a decade or so ago). In other words, if

you had a project in which com

putational approaches could play an important

part—in generating or testing hypotheses on the basis of a large body of data, for in-

stance—then you w

ould call in the services of a specialist computational m

usicolo-gist, w

ho would arrange for any necessary encoding of data, carry out the analysis,

and provide advice on the interpretation of the results. What the ordinary m

usicolo-gist w

ould need to know, then, w

ould be not how to extract or collapse H

umdrum

spines, but how to form

ulate a research question in such a way that advantage can

be taken of computational m

ethods to answer it m

ore securely than is possible usingthe m

ethods traditional in data-poor fields. Whether anyone could m

ake a decentliving as a specialist com

putational musicologist is another m

atter.U

sing computers for m

usicology is like using computers for anything else: you

need to have reasonable expectations about what they can do. H

owever open and

flexible its design, any musicological softw

are is a tool, and like any tool it is goodfor som

e things and not for others. The articles by H

uron and his coworkers that I

have described in this chapter might be characterized as finding m

usicological usesto w

hich computational tools can be put; in essence, they illustrate the softw

are’s po-tential, and they do so in a m

anner that owes m

ore to social-scientific discourse thanto traditional hum

anities writing, w

ith their explicit hypotheses, control groups,quantification of significance, and form

ulation in what H

uron (1999b) has termed

the “boilerplate language” of scientific inquiry. Perhaps that is only to underline thedistinction betw

een “cognitive” or “systematic” m

usicology, as such work is som

etimes

called, and the close attention to context and what m

ight be termed epistem

ologicalpluralism

of the traditional discipline. But what I am

suggesting is that musicology in

the broadest sense can take advantage of computational m

ethods and transform itself

into a data-rich discipline, without giving up on its hum

anist values. Understood that

way, scientific discourse is unlikely to offer a general m

odel for the musicology of the

future: it is up to musicologists to develop such a m

odel. And that does not m

ean try-ing to find m

usicological uses to which com

putational tools can be put, but the re-verse: discovering the tools that are m

ost appropriate to a particular musicological

purpose, and taking full advantage of them. A

t present we hardly have such a thing as

a model of how

sophisticated computational approaches m

ight be integrated within

a sustained, musicologically driven project. Yet the tools are ready and w

aiting. 20

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parative Musicology

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Notes

1.T

he routines used to create these graphs, which took advantage of the N

eXT

com-

puter’s Postscript display, have not to date been disseminated, but the principles

underlying them w

ere described in chapter 20 of Brinkman 1990.

2.H

elmut Schaffrath, Essen M

usical Data Package

[Menlo Park, C

alif.: Center for C

om-

puter Assisted R

esearch in the Hum

anities (CC

AR

H), 1995] [data and analytical

software on four floppy discs for M

S-DO

S]); 6,225 folksongs are included in thisrelease. Both data and softw

are are now available at w

ww

.esac-data.org. (All U

RLs

were correct at press tim

e, but Web resources change rapidly. Search engines m

ayprovide a better m

eans of access to the resources listed in this chapter.3.

A full listing of the analytical output for “D

eutschland über alles,” reformatted and

annotated for clarity, may be found in Selfridge-Field 1995; see also Schaffrath

1992, 1997.4.

Since EsAC

consists entirely of ASC

II characters, it is possible to import the records

into modern databases. H

owever researchers now

generally prefer to access the kernversion of the databases and process them

using Hum

drum (see note 7 below

).5.

Hence the need for translation program

s such as POC

O (see chapter 8, this volum

e).6.

Digital A

lternate Representation of M

usical Scores; for details see Selfridge-Field1997, chapters 11

–15.

7.D

avid Huron, T

he Hum

drum Toolkit: Softw

are for Music R

esearchers[Stanford, C

alif.:C

enter for Com

puter Assisted R

esearch in the Hum

anities, 1993 (three floppy discsand 16-page installation guide)]. A

wide range of inform

ation may be accessed

from the H

umdrum

Toolkit home page (w

ww

.humdrum

.net) and from the m

ainH

umdrum

site (ww

w.m

usic-cog.ohio-state.edu/H

umdrum

or http://dactyl.som.ohio-state.edu

/Hum

drum).

8.U

NIX

is an operating system, like M

S-DO

S or Window

s, but includes a large num-

ber of general-purpose tools and thus fulfils many of the functions of a program

ming

environment as w

ell. Developed in the late 1960s, it rem

ains the environment of

choice for many professional softw

are developers and researchers.9.

For a summ

ary of kernsee H

uron 1997; for detailed descriptions see Huron 1995,

1999a.10.

Lines beginning with asterisks are know

n as “interpretation records,” those with tw

oasterisks being inclusive

interpretations (something is either in kern or it isn’t), and

those with one asterisk being tandem

interpretations (of which you can have any

number).

11.In a review

of the Hum

drum Toolkit, Jonathan W

ild (1996) includes what is essen-

tially a tutorial on this precise task; assuming that the chorales have been concate-

nated into one file, the entire analysis consists of seven comm

ands pipelined toanother—

in other words, it can be executed w

ith a single comm

and line. Furtheranalytical applications of H

umdrum

are discussed in Huron 2002, w

hich appearedafter this chapter had gone into production.

12.See chapter 1, this volum

e.13.

A follow

-up study (Hippel 2000) reanalyzed the experim

ental data on the basis ofw

hich Burton Rosner and Leonard M

eyer had claimed that gap-fill patterns reflect

listener expectations; von Hippel concludes that that the apparent effects of gap-fill

expectations were an artifact of experim

ental design (specifically, effects of training).O

f course listeners may have such expectations, but von H

ippel’s and Huron’s point

is that these are the result of melodic patterns and not the other w

ay around.

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14.See the FA

Q page at the H

umdrum

Toolkit home page (http://dactyl.som

.ohio-state.edu/H

umdrum

/index.html). A

dditional tools developed by Craig Sapp

can be downloaded from

http://ww

w.ccarh.org

/software/hum

drum/m

useinfo.15.

There is a further alternative: a num

ber of Hum

drum com

mands can be run online

at http://musedata.stanford.edu

/software/hum

drum/online; you select the com

-m

and you want, supply the input data (by pasting it into a w

indow, uploading a file,

or supplying its UR

L), set the options, and receive the output as plain text or inH

TM

L. Twenty of the 70 or m

ore Hum

drum com

mands are currently available in

this way, w

ith an emphasis on data translation and conversion.

16.T

here are two principal sources for such m

aterial (from both of w

hich they may be

obtained free): the KernScores site at http://kern.hum

drum.net, and C

CA

RH

’sM

useData site (http://w

ww

.musedata.org).

17.“H

umdrum

Toolkit GU

I” currently exists only in a 16-bit version, running underW

indows 3.1, but further developm

ent (and eventual distribution) is planned by theSonic A

rts Centre at Q

ueen’s University, Belfast.

18.O

nly a few of these “regular expressions,” as they are know

n in UN

IX jargon, are

built into the interface, but you can add your own, along w

ith new interpretations;

this is where you w

ould define **timing or **heartbeat.

19.A

ccessible at http://ww

w.them

efinder.org; for a brief description see Kornstädt

1998.20.

My grateful thanks to D

avid Huron, w

ho has influenced this chapter in many w

ays,not least through m

aking it possible for me to w

ork with him

in 2000 as a visitingscholar at the C

ognitive Musicology Laboratory, O

hio State University, C

olumbus.

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