From a Microtonal Scale to a New Musical Grammar

This dissertation addresses the problem of
musical knowledge acquisition by investigating humans' learning
of a novel and unfamiliar musical system. This chapter details
the novel musical system, including how and why it is used in
composition and in the following experiments.

The Bohlen-Pierce Scale

In most musical systems of the
world, frequencies of notes are based around an octave, which
is defined as a 2:1 ratio in frequency. Within the octave, the
equal-tempered Western musical scale is logarithmically divided
into 12 even increments. Thus, the frequencies of tones
spanning an octave in the Western scale are defined as the
following:

Frequency (Hz) =
k*2n/12

where n is the number of
steps along the chromatic scale, and k is a constant
which specifies the starting frequency of the scale; e.g. 440Hz
or A4, the most common tuning pitch in modern orchestras.

For the following experiments
we used the Bohlen-Pierce scale (Mathews, 1988; Sethares, 2004;
Walker, 2001), a microtonal tuning system based on 13
logarithmically even divisions of a tritave, which is a
3:1 frequency ratio. The tones in one tritave of the
Bohlen-Pierce scale are defined as:

Frequency (Hz) =
k*3n/13

where n is the number of
steps along the tritave scale, and k is a constant,
defined here as 220Hz. The choice of 220Hz as the starting
point of our musical system allows for steps along the first
tritave to span the frequency range of the middle octave of the
piano; thus the frequencies derived from the formula fall
within the optimal range of musical pitch (Attneave &
Olson, 1971; Huron, 2001).

From the Bohlen-Pierce scale
formula we defined the following pitches in one tritave of the
Bohlen-Pierce scale (see Figure 1):

Figure 1. Frequencies along the
Bohlen-Pierce scale and Western scale.

The Bohlen-Pierce scale was
first derived mathematically by H. Bohlen and J. Pierce in the
1950s. Since then it has been used in a number of contemporary
compositions such as several pieces by Boulanger (e.g. "I Know
of No Geometry") and Blackwood's Microtonal Etudes
(2004). Chords in the B-P scale were derived mathematically in
Krumhansl (1987), and were tested in consonance judgments by
Mathews et al (1988). The unique interaction between harmony
and timbre in the B-P scale was explored in Sethares's book
Tuning, Timbre, Spectrum, Scale (2004). An example of
Pachelbel's Canon tuned in the equal-tempered Bohlen-Pierce
scale can be heard here: [http://www.io.com/~hmiller/midi/canonBP-tempered.mid]
(Source: [http://members.aol.com/bpsite/])

Although there has been a
substantial amount of work on the Bohlen-Pierce scale from both
theoretical and compositional perspectives, studies of music
cognition have yet to address musical systems using this scale.
The Bohlen-Pierce scale offers a novel way to approach the
study of how the human brain acquires music; however, to be
able to address these questions scientifically, individual
compositions in the Bohlen-Pierce scale are not enough. There
must be a scientifically rigorous definition of how
experimental stimuli are composed, that will convert a
mathematically derived scale used in avant-garde music into
well-defined stimuli for use in controlled studies. This
chapter proceeds to describe the derivation of such a new
musical system.

Chord Progressions in the Bohlen-Pierce
Scale

Based on the Bohlen-Pierce
scale formula with 220Hz as the initial reference tone:

Frequency (Hz) = 220 * 3 ^
(n/13),

it was possible to define
chords using pitches with frequencies that approximately relate
to each other in low-number integer ratios, which are perceived
as being relatively consonant psychoacoustically (Kameoka &
Kuriyagawa, 1969). One "major" chord in this new system is
defined as a set of three pitches with frequencies that
approximate a 3:5:7 ratio (see Figure 2; also see Krumhansl,
1987, for a derivation of chords in the Bohlen-Pierce
scale).

Figure 2. Choosing pitches that
form a 3:5:7 harmonic ratios in frequency to form chords.

By combining pitches in the
Bohlen-Pierce scale to form chords, and then stringing together
chords to form chord progressions, many sets of chord
progressions can be formed. We composed two sets of chord
progressions, where each chord progression consisted of four
chords, and each chord consisted of three pitches. Table 1
lists each of the pitches in each chord, with the frequency of
each pitch given by substituting the pitch number into n
in the Bohlen-Pierce scale formula: Frequency (Hz) = 220 *
3n/13.

Pitch number

Grammar I

Grammar II

Table 1. Pitches in each chord
progression, where each number represents a pitch, each column
of numerals represents a chord, and each 4*3 matrix of numbers
represents a chord progression. Each number refers to "n" in
the formula for frequency in the B-P scale: F =
220*3n/13.

These chord progressions constitute the
underlying structure of two musical grammars which we use in
the present study.

Composing Melodies from Chord Progressions -
Applying Finite-State Grammars

The chord progressions formed the
underlying grammatical structure of all stimuli used in the
experiments to be reported in the next few chapters. To compose
melodies in the Bohlen-Pierce scale, we applied a finite-state
grammar, a process which allows the generation of grammatical
items from an underlying structure.

Grammars are sets of rules that specify
how items (such as letters, syllables, words, or in this case,
musical notes) could be arranged together. A typical
finite-state grammar (e.g. Reber, 1989) consists of a set of
nodes representing the state, with one or more possible items
contained in each state; these nodes are connected by links
representing possible pathways between two or more items. In
the finite-state grammars used in this dissertation, each chord
was treated as a state, and each pitch as a possible node in
the state. Each melody must begin at the first chord and end at
the last chord. Each pitch in the melody must belong to a
chord, and each pitch must either stay on the same chord or
move forward to a pitch in the next chord. Thus the
finite-state grammar can be illustrated as a lattice of
possible pathways connecting all the pitches as nodes. To
compose a melody in one of the two grammars, one begins by
selecting a first note from among the three possible notes in
the first chord; then the next note is selected by following
any of the five possible pathways that either lead to a
repeated note, a note up or down within the same chord, or a
move forward to one of the three notes in the next chord.
Figure 3 illustrates the finite-state grammar that is derived
from the lattice of numbers that constituted all the pitches in
the chord progressions, and Figure 4 illustrates one possible
melody composed from one of the two grammars.

Fig. 3. A finite-state grammar diagram
illustrating the composition of melody from harmony. Legal
paths are shown by the arrows.

Figure 4. An illustration a melody
based on finite-state Grammar 1. Dark arrows illustrate the
paths taken, whereas light arrows illustrate the other possible
paths that are legal in the grammar. The resulting melody is
shown at the bottom of the figure.

This artificial musical system adheres
to many of Lerdahl and Jackendoff's rules for a generative
theory of tonal music (Lerdahl & Jackendoff, 1983). For
example, the musical grammars are parsable, groupable,
hierarchical, and statistically predictable. This makes the
system viable as a new compositional tool, yet capable of
generating melodies that are completely unfamiliar to all
participants in our studies. Using these finite-state grammars,
a large number of melodies can be generated (3 ^ 8 = 6561
possible melodies). A small subset of these melodies are legal
exemplars of both Grammar I and Grammar II. These melodies were
included in the exposure phase, but were not used as test
items.

Having defined a true (albeit
constrained) musical grammar, we now have a system with which
to assess several aspects of learning. These musical grammars
support important studies not only in music cognition, but also
in statistical learning more generally. In particular, the
materials constituting the finite state grammars differ from
typical grammars (e.g. Reber, 1967) in that the items
themselves (i.e. tones) do not readily lend themselves to
explicitly designated labels such as letter names. Thus one is
forced to form a non-verbalizable mental representation of
these tones. Unlike existing artificial grammar studies e.g.
those using artificial speech syllables, this system forces the
mind to represent items in a non-linguistic manner, thus
minimizing the possibility of rote memorizing the items of an
artificial grammar through covert verbalization of its
exemplars. Thus, the new musical system is useful as a tool to
assess the constraints of statistical learning by testing the
domain-generality of the learning mechanism (Saffran,
2002).

Specific Research Questions

The next few chapters will report
studies investigating several aspects of learning in the new
musical system:

1. whether human
subjects are sensitive to the relative frequencies of tones
perceived; (Chapter 3)

2. what brain
mechanisms underlie the formation of musical expectancies
(Chapter 5);

3. whether humans are
able to learn the underlying rules that constitute a
finite-state grammar; (Chapter 3)

4. whether exposure to
these novel stimuli can alter the human emotional response to
melodies; (Chapters 3 and 4)

5. whether the level
of prior expertise in an existing musical system will aid in
the learning of the new musical system (Chapters 4 and 5);

6. and whether the
manipulation of musical and psychoacoustic factors such as
timbre, harmony, and melodic intervals influence learning of
the musical system (Chapter 4).

Having outlined our new musical
system and the different rules used to make a comprehensive
musical grammar, the next chapter will present the original
findings showing that humans can, in fact, demonstrate some
sensitivity to underlying statistical structure (relative
frequencies) in a new musical system, as well as develop
preference for melodies that are familiar. Chapter 4 will
extend the findings of Chapter 3 by further characterizing the
conditions under which our new musical context can best be
learned. Chapter 5 will provide physiological evidence both as
a convergent method, and to link the behavioral findings into a
broader theoretical context.