ALGORITHMIC STOCHASTIC MUSIC by ALEX COOKE

49 Pages·2016·4.59 MB·English

ALGORITHMIC STOCHASTIC MUSIC

by

ALEX COOKE

Submitted in partial fulfillment of the requirements for the degree of Master of
Science

Departments of Mathematics, Applied Mathematics, and Statistics

CASE WESTERN RESERVE UNIVERSITY

May 2017
CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the thesis/dissertation of

Alex Cooke

Candidate for the degree of Master of Science

Committee Chair

Wojbor Woyczynski

Committee Member

Wanda Strychalski

Committee Member

David Gurarie

Date of Defense

4/5/2016

*We also certify that written approval has been obtained for any proprietary
material contained therein.
4.3 Bibliography 47
List of Figures
1 The C Major scale. . . . . . . . . . . . . . . . . . . . . . . . . 4
2 The a minor scale, relative minor of C Major. . . . . . . . . . 4
3 The triads in C Major: C major, d minor, e minor, F major,
G major, a minor, and b diminished. . . . . . . . . . . . . . . 5
4 The movement between functions. . . . . . . . . . . . . . . . . 5
5 Some typical harmonic progressions. . . . . . . . . . . . . . . . 8
6 An example melody. . . . . . . . . . . . . . . . . . . . . . . . 9
7 A sample realization. . . . . . . . . . . . . . . . . . . . . . . . 11
8 A modulation via a secondary dominant. . . . . . . . . . . . . 12
9 Modulation to new random walk. . . . . . . . . . . . . . . . . 13
10 A stochastic matrix of frequency and intensity, slices of time
with corresponding frequencies, and a sample screen and its
evolution in time. . . . . . . . . . . . . . . . . . . . . . . . . 16
11 The musical realization. Note that slots two and three are
combined into one note; this provides some rhythmic variation. 29
12 The musical realization. . . . . . . . . . . . . . . . . . . . . . 32
13 The pitch set {0,1,4,6,7}. . . . . . . . . . . . . . . . . . . . . 40
14 Pitch set harmonies. . . . . . . . . . . . . . . . . . . . . . . . 41
15 The minor heat map and an example of re-weighting it to
emphasize a somber mood. . . . . . . . . . . . . . . . . . . . . 46
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Algorithmic Stochastic Music

Abstract

by

ALEX COOKE

The aim of this project is to develop a system of tonal musical
generation that can recognize shifts in tonal center via random
processes and reorient itself accordingly. One might liken this
to a random walk with the added property of boundary
crossings. These weights can be conveniently encoded in the
discrete probability mass functions corresponding to each
scale degree. The result is a self-generating system of music
that follow tonal rules in a convincing way.
1 Introduction
1.1 Musical Motivations and Background
Tonal music is by far the most recognizable form of Western music to the
majority of musicians and non-musicians alike. From pop music to a large
portion of classical repertoire, the majority of music is built upon or deriva-
tive of the tonal system, which uses an asymmetric organization of a series
of pitches to induce relationships based on movements between three types
of triadic harmony (harmonies built on the interval of a third).
A key is a collection of pitches around a certain pitch, based on the order-
ing set forth by a major or minor scale. A major scale is given by the se-
quence of whole steps (two semitones, or adjacent pitches) and half steps
(one semitone): WWHWWWH, whereas a minor scale is given by: WH-
WWHWW. Note that a minor scale is equivalent to a major scale begin-
ning on the sixth scale degree. Such a relationship is called the ”relative”
key.
Figure 1: The C Major scale.
Figure 2: The a minor scale, relative minor of C Major.
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By stacking notes together that are a third apart, one creates ”triads”;
these harmonies form the basis of motion in music. Triads are named based
on their root (the note of the scale upon which two other notes are stacked)
and their ”quality,” the size of the steps between notes. A major third
(four semitones) followed by a minor third (three semitones) is a major
triad, a minor third followed by a major third is a minor triad, and two
minor thirds form a diminished triad.
Figure 3: The triads in C Major: C major, d minor, e minor, F major, G
major, a minor, and b diminished.
The three basic categories of harmonies are tonic, predominant, and dom-
inant, so named because of their respective functions. Whereas chords can
be augmented by other tones, substituted for one another, or borrowed
from other keys, their individual functions can all be placed in one of these
three categories, within which certain rules govern their movement within
and between the categories.
Figure 4: The movement between functions.
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Tonic is the most the versatile of the functions and is typically thought of
as the ”home” of a key. Harmonies that belong to the tonic category are
either the chord built on the root of the key, or share two of their three
notes with the chord built upon the root (called tonic substitutes). Pre-
dominant chords serve as an intermediary for tonic and dominant chords;
they overwhelmingly move to the dominant. Rarely, they move back to the
tonic in a motion called a ”plagal cadence.” This is both rare and highly
specific and does not drastically alter the function of a predominant har-
mony. Dominant chords serve to move back to tonic to reinforce it as the
base of a key. They accomplish this through two main mechanisms: a de-
scending fifth motion, considered the strongest relationship in tonality, and
the half-step relationship between the seventh scale degree and the tonic
degree, which creates a strong pull toward the tonic, so strong that it is
called the ”leading tone.”
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In major, triads are categorized as follows:
Scale Degree Triad Quality Function
I Major Tonic
ii minor Predominant
iii minor Tonic
IV Major Predominant
V Major Dominant
vi minor Tonic
vii° diminished Dominant
In minor, triads are categorized as follows:
Scale Degree Triad Quality Function
i minor Tonic
ii° diminished Predominant
III Major Tonic
iv minor Predominant
v minor Dominant
VI minor Tonic
VII Major Dominant
One may stay in a category as long as one pleases, but when one moves be-
tween categories, the rules of Figure 4 must generally be followed. There
are other rules of good harmonic practice that dictate the motion between
individual harmonies (namely, the vertical arrangement of the harmony,
known as ”voice leading”) that are beyond the scope of this thesis; they
will often be simplified or overlooked for the sake of illustration. Some typ-
ical harmonic progressions are shown next:
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Figure 5: Some typical harmonic progressions.
The first progression is the most basic harmonic motion: I-V-I (T-D-T).
The second progression introduces a very typical predominant chord, the
IV chord (also known as the subdominant). The third example illustrates
the same progression in minor. Very often, the v chord in minor will be
replaced with the V chord from the parallel major (the major key sharing
the same tonic), as the this replaces the seventh scale degree (the third of
the chord) with the major seventh scale degree, which is a half-step from
tonic, creating a stronger tendency to resolve. So often is this done that it
is not often labeled as a borrowed chord. The final chord illustrates a ”de-
ceptive” cadence, a movement from V to vi, which subverts the expectation
of a V-I movement. Such a movement could be used to modulate (intro-
duce a new key) to the relative minor, which would make vi the new i.
While harmonies provide the basis of motion within a piece, melodies can
be thought of as the ”singable” part of music. They are typically single
note lines that consist of notes taken from the underlying harmony; whereas
harmony is the vertical component, melody is the horizontal. The contour
of a melody may also be improved by the use of non-harmonic tones that
make for a smoother line or create musical tension and resolution. Such
non-harmonic tones may include steps (passing tones), neighbors (a move
to a note directly above or below and back), anticipations (a sounding of a
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