INFORMATION CONTENT IN MELODIC AND NON-MELODIC LINES

Ben Duane

Northwestern University

ABSTRACT

Although melodies seem to be focal points for attentionfigures to

the grounds created by other musical lineslittle is known about

why this is so. This paper tests the hypothesis that melodies mark

themselves for attention partly by being less predictable than the

lines that accompany them. As in several previous studies,

predictability is quantified using various types of information

entropy. These entropies are computed for melodic and

non-melodic lines extracted from two musical corporaone

containing rock songs, the other containing Baroque keyboard

minuets. Results show that the various entropies not only are

significantly higher in melodies than in non-melodies, but can be

used to classify lines as melodic or non-melodic with above-chance

accuracy.

1. INTRODUCTION

Melodies, more than other musical lines, seem to attract listeners

attentionto occupy the foreground of their perception. Indeed,

the whole concept of melody is often explained using the

figure-ground metaphor. Melodies are not just coherent series of

notes, they are principal. They stand apart from their

accompaniments, like faces from the backgrounds of portraits, and

place themselves under the listeners notice.

Figure 1 Boccherini, string quartet, op. 11, no. 5, mm. 1-4.

When listening to Figure 1, for instance, ones attention is drawn

more to the melodic line of the Violin 1 than to the non-melodic

lines of the other four parts. It is for this reason, I suspect, that

Boccherinis minuet can be identified easily by the first violins line

but not by one of the other four. For someone who did not know the

piece by name, it would make perfect sense to refer to the quintet

whose melody goes:

But not the quintet with that cello line:

The Violin 1 melody is a signature of sortsa feature by which the

piece can be instantly recognizedand it could hardly function this

way without being a focal point of the listeners attention.

This much is almost common sense. Most of us realize, at least

intuitively, that melodies are uniquely marked for attention. But

what marks them? What about melodies makes them stand out,

attract attention, acquire principality? One possibility is that our

attention is drawn not to the melody per se but to the top of the

texture, since the melody is usually the highest line present. This,

however, does not explain excerpts like the Boccherini, in which

the non-melodic Violin 2 and Viola are often above the melodic

Violin 1. The effect might also be due not to the melody but its

performance. One might, in other words, attend to Figure 1s first

violin part because the first violinist brings it out. But how would

he or she know to do this? First violinists cannot count on always

having the melody. No, it must be something about the line itself,

not how it is played or orchestrated, that designates it as melody and

prompts the listener to attend to it.

That something, which marks melodies for attention, might be

partly their predictabilityor lack thereof. In Figure 1, the Violin 1

line is, in every apparent way, less predictable than the other four.

This line features nothing as pervasive and uniform as the second

violins Es, the violas eighth notes, the first cellos thirds, or the second cellos articulations. The melody is simply the least

predictable, most interesting line in this example. And so it is in

many other examples from many other musical styles. Countless

accompanimental patterns, from the Alberti bass to the oompah

bass, are more predictable than the melodies they are paired with.

Perhaps, then, the lower predictability of melodies is operativea

feature that somehow directs our attention toward them.

This hypothesis is tested here. Like several previous authors, I

quantify the predictability of musical lines using different types of

information entropy. These quantities are calculated across two

musical corpora, each containing one melodic line and one

non-melodic line. The results suggest that melodies indeed tend to

be less predictable than non-melodies, meaning that this lower

predictability might mark them for attention.

2. RELATED RESEARCH

Madsen and Widmer (2006) tested essentially the same hypothesis

as this paperthat listeners tend to focus on the most complex

(least repetitive) voice, experiencing this as foreground (p. 1812).

Like me, these authors quantify complexity using different types of

information entropy, but their work differs from mine on two

counts. First, they compute some types of entropy that I do not, and

vice-versa. (They do not compute the sub-phrase entropy described

in section 3.4, for instance.) Second, their data comprise one

symphony and one concerto, whereas mine include a corpus of

minuets and a corpus of rock songs.

Although psychologists have not, to my knowledge, studied why

melodies attract attention, they have researched musical attention

from other angles. Davison and Banks (2003) had subjects listen to

two-voice counterpoint, instructing them to attend to one of the

voices, and found that their perception of the voice attended to was

affected by the structure of the voice ignored. Dowling, Lung, and

Herrbold (1987) found that if listeners are primed to expect notes in

certain pitch regions, and at certain times, they often direct their

attention accordingly. And Bigand, McAdams, and Fort (2000)

tested two competing models of musical attention: a divided

attention model, in which listeners concurrently attend to multiple

lines; and a figure-ground model, in which listeners attend to just

one part at a time. The authors results, however, did not fully

support either hypothesis, leading them to propose a third,

integrative model, by which listeners switch their attention

between one voice at a time and all voices at once.

Several researchers have applied information theory to the study of

music. Meyer (1967a, 1967b) discussed connections between

information theory and musical meaning and aesthetics. Other

authors have tried to use information entropy as a metric of musical

style (Youngblood 1958, Knopoff and Hutchinson 1983, Snyder

1990, Margulis and Beatty 2008). Hiller (1967) performed an

information-theoretic analysis on Weberns Symphonie, op. 21.

And Knopoff and Hutchinson (1981) proposed a method for

computing information-theoretic quantities for musical continua.

None of this research, however, attempted to use entropy to predict

whether lines are melodic or not.

3. QUANTIFYING PREDICTABILTY

3.1 First- and Second-Order Entropy

The central premise of information theory is, of course, that

information and predictability are inversely relatedthat the

unexpected is also the most informative. Mathematically, this

means that as probability decreases, information increases or, more

specifically, that the information, I(x), of some event x is given by

Equation 1:

2logI x p x (1)

where p(x) is the probability of x.

Entropy, then, is the average information of each event in a signal

(i.e. a series of events). The first-order entropy, H(1), of a signal is

defined as:

1 1

21log

n

i ii

n

i ii

H p x I x

p x p x

(2)

where n is the number of possible events xi. (That first-order

entropy derives from zeroth-order Markov probability is an

unfortunate quirk of standard terminology.) Equation 2 is

conceived as a mean of the information, I(xi), across all possible

events xi, weighted by the respective probabilities, p(xi). As such,

this mean estimates the expected information of each event in the

signal or, in another sense, each events average surprisal.

Say, for example, that the signal is a series of coin flips. The

alphabet of possible events would be X = {heads, tails}, and the

first-order entropy of the signal would be:

2 21 log logH p heads p heads p tails p tails (3)

If the coin was fair, and p(heads) = p(tails) = 0.5, the entropy would

be H(1) = 1. But if the coin was riggedsay with p(heads) = 0.75

and p(tails) = 0.25then the entropy would be H(1) = 0.81. Entropy

is lower with the rigged coin because that coin is more predictable:

most of the time, it will turn up heads, whereas the first coins

distribution is half-and-half.

Second-order entropy is computed using first-order transition

probabilitiesthat is, the probabilities of observing certain events,

given the events directly preceding them. The formula is:

22 1 1 log

n n

i i j i ji jH p x p x p x

(4)

where p(xi) is the probability of event xi and pi(xj) is the probability

of event xj given that event xi has just occurred. Equation 4 could be

rewritten as:

2 11

n

i iiH p x H x

(5)

where H(1)(xi) is the first-order entropy of the signal, given that xi

was the most recent event. Second-order entropy, then, is a mean of

the first-order entropies H(1)(xi) for each prior event xi, weighted by

the probabilities p(xi) of these events.

3.2 Pitch Entropy

In what follows, pitch entropy is defined using the alphabet of

possible melodic intervals X = {leap up, step up, unison, step down,

leap down}. To compute this entropy for a given line, each interval

is placed into one of the categories in X, as illustrated in Figure 2.

Each intervals probability, p(x) for each xX, is estimated by counting that intervals instantiations. The lines entropy is then

calculated from these probabilities.

Figure 2 Vocal line from She Loves You by the Beatles.

We could, of course, expand the alphabet of intervals. The leap up

category, for instance, could distinguish between thirds, fourths,

fifths, and so on. But larger alphabets can become too large for the

relatively short signals found in music. If an alphabet contains n

intervals, then as n increases, so does the likelihood that some

intervals will be absent from a given musical line. Such absences

become even more likely with the n2 possible pairs of intervals

reflected by second-order entropy. And, when many possibilities

are not represented, entropy seems to become less effective in

distinguishing musical signals. The size of the interval alphabet is

kept low for this reason.

3.3 Rhythmic Entropy

Entropy is also computed with respect to rhythmic durations. For

each note in the line under analysis, the following value is obtained

(see Figure 2):

2logb round b (6)

where b is the number of beats the note occupies, and round{}

returns the integer nearest . Rhythmic entropy, then, is defined

using the alphabet X = {b < 3, b = 3, b = 2, b = 1, b = 0, b = 1, b = 2, b = 3, b > 3}. This definition serves the dual purpose of transforming rhythmic duration from a continuous to a discrete

variable, which makes entropy simpler to calculate, and limiting the

number of possibilities, for the reason discussed in section 3.2.

3.4 Entropy of Sub-Phrases

We seem, as listeners, to process melodies and other lines not only

as complete units but also as series of shorter segments. (The

melody in Figure 2, for example, is easily heard as the segments

shown beneath the staff.) It is plausible, therefore, that computing

the entropy of such segments, rather than of full lines, would be

usefulthat it would tap into some key aspect of music perception

that we would otherwise overlook. I obtained such quantities for the

analysis reported below. Melodies and other lines were

automatically segmented into sub-phrases, using an algorithm

proposed by Cambouropoulos (1997), and then entropy was

computed for these sub-phrases.

4. MUSICAL CORPORA

To assess whether melodic lines are in fact less predictable than

non-melodic lines, two corpora of music were assembled. Two

linesone melodic, the other non-melodicwere extracted from

each musical work. For each line, six varieties of entropy were

computed: first- and second-order pitch and rhythmic entropy for

the complete lines and first-order pitch and rhythmic entropy for

lines segmented into sub-phrases.

The first corpus included the nine two-part minuets from J. S.

Bachs Notebook for Anna Magdalena Bach. In this case, the right

hand was considered melodic, the left hand non-melodic. The

second corpus comprised the first verse and chorus from thirteen

rock songs by The Beatles, Cream, Tom Petty, Billy Joel, and Elvis

Costello. For each song, I transcribed the vocal line, which was

considered melodic, and the bass line, which was considered

non-melodic.

5. RESULTS

The data from each corpus were analyzed in two ways. Multivariate

analyses of variance and paired t-tests were used to compare the

entropies of melodic and non-melodic lines. And linear

discriminant analysis (LDA), quadratic discriminant analysis

(QDA), and logistic regression (LR) were employed to see how

well the six entropies could classify lines as melodic or

non-melodic. I tested classifiers using not only all six entropies

together, but also the other sixty-two possible subsets of these six.

Accuracy was defined as the percentage of pieces in the corpus that

were correctly classified, and statistical significance was evaluated

using a binomial test. The classifiers were tested using

cross-validation. For each piece, a classifier was trained on the

other members of the corpus, and this classifier was tested on the

withheld piece.

The results for the minuets are shown in Tables 1 and 2. A

MANOVA showed a significant difference in one dimension

between the entropies of the right and left hands (p = 0.01).

Follow-up t-tests showed that both second-order pitch entropy for

full lines and first-order pitch entropy for sub-phrases were

significantly higher in melodic than in non-melodic lines. None of

the other entropies produced significant differences. The LDA

classifier performed above chance when using all six entropies,

labeling the minuets with 83.33% accuracy (p < 0.01). Several

subsets of the six entropies increased the accuracy of all three

classifiers to 88.89% (p < 0.001).

ENTROPY RIGHT HAND LEFT HAND

P-VALUE Mean St. Dev. Mean St. Dev.

1st-order,

pitch

1.51 0.22 1.26 0.21 0.10

1st-order,

rhythmic

2.02 0.12 1.97 0.18 0.73

2nd-order,

pitch

1.11 0.15 0.85 0.15 0.01

2nd-order,

rhythmic

1.54 0.15 1.59 0.15 1.00

1st-order,

pitch,

sub-phrases

1.50 0.22 1.09 0.16 0.01

1st-order,

rhythmic,

sub-phrases

1.98 0.18 1.82 0.25 0.20

Table 1 Distributions of entropies for the minuets.

CLASSIFIER ALL SIX ENTROPIES BEST SUBSET(S)

Accuracy p-value Accuracy p-value

LDA 83.33% 0.004 88.89%

rhythmic for sub-phrases, were significantly higher in vocal lines

than bass lines. All three classifiers performed above chance when

using the six entropies. Each classifiers performance improved

with certain subsets of the entropies, with the LR model reaching

100% accuracy in one case.

ENTROPY VOCAL LINE BASS LINE

P-VALUE Mean St. Dev. Mean St. Dev.

1st-order,

pitch

1.85 0.29 1.20 0.51