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이학석사 학위논문

Computational Analysis of

Key and Harmony

focusing on Bach’s Works

(바흐의 작품을 중심으로

조와 화성에 대한 계산적 분석)

2018년 2월

서울대학교 대학원

수리과학부

권 영 흠

Abstract

Youngheum Kwon

Department of Mathematical Science

The Graduate School

Seoul National University

This study aimed to analyze key and harmony in music. Works by Johann Sebastian Bach are the

main material of this study. Using works by Bach, the main target of analysis are three properties of

keys. Major and minor is to observe differences of major keys and minor keys. Key characteristic

is about characteristics of each major and minor key. Key distance is to check theoretical distance

between keys.

Temperaments determine frequencies of pitches and affect harmony. Frequency ratios are the

fundamental part to compute harmony in this study. The ratios were intentionally designed to analyze

harmony in musical works and to check three properties of keys. Using them, vertical harmony (VH)

and horizontal harmony (HH) were computed for each musical work, and their dispersion within a

work was also defined and computed.

From the result of computation, this study answered three properties of keys. Major and minor

was verified so that harmony in minor is 35% (47%) more intensive (more dispersed) then harmony

in major. Key characteristic was too weak to be significant for both major and minor keys. Key

distance was not realistic, being unsupported by computation.

Keywords : Bach, Temperament, Frequency ratio, Key, Harmony

Student Number : 2015-20252

Contents

1 Introduction 1

1.1 Analysis of musical works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Semitones and octaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Temperaments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Keys and the circle of fifths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Properties of keys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Foundation 10

2.1 Classical theory of consonance and dissonance . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Wave and frequency ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Interval Harmony (IH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Modeling 19

3.1 Pitches to Pitch Value (PV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 PV to ID to IH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Vertical Harmony (VH) and Horizontal Harmony (HH) . . . . . . . . . . . . . . . . . . 20

3.4 Mean and sum of VH and HH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.5 Initial Rhythm Set (IRS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.6 Dispersion of VH and HH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Computer process 25

4.1 Score to pitches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Pitches to IH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Intensity factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.4 Dispersion of the four factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Analysis 30

5.1 Major and minor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Key characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3 Key distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

List of Figures

1.1 The circle of fifths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 BoK=25, LoFE=33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 BoK=31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 BoK=31, LoFE=16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Prelude in BWV 847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Prelude in BWV 847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Prelude in BWV 847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 An algorithm of PV to ID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.4 Prelude in BWV 847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.5 Prelude in BWV 846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.6 Prelude in BWV 846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.7 VH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.8 HH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 Bach, Major . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Bach, minor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3 Bach, Major and minor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.4 Major, Intensity factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . . . . 33

5.5 Major, Intensity factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.6 Major, Intensity factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . . . . 33

5.7 Major, Dispersion factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . . . 34

5.8 Major, Dispersion factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.9 Major, Dispersion factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . . . 34

5.10 minor, Intensity factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . . . . . 35

5.11 minor, Intensity factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.12 minor, Intensity factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . . . . 35

5.13 minor, Dispersion factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . . . . 36

LIST OF FIGURES LIST OF FIGURES

5.14 minor, Dispersion factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.15 minor, Dispersion factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . . . 36

5.16 Major, VH Intensity factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . . 38

5.17 Major, VH Intensity factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.18 Major, VH Intensity factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . . 38

5.19 Major, HH Intensity factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . . 39

5.20 Major, HH Intensity factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.21 Major, HH Intensity factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . . 39

5.22 Major, VH Dispersion factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . 40

5.23 Major, VH Dispersion factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . 40

5.24 Major, VH Dispersion factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . 40

5.25 Major, HH Dispersion factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . 41

5.26 Major, HH Dispersion factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . 41

5.27 Major, HH Dispersion factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . 41

5.28 minor, VH Intensity factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . . 42

5.29 minor, VH Intensity factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.30 minor, VH Intensity factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . . 42

5.31 minor, HH Intensity factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . . 43

5.32 minor, HH Intensity factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.33 minor, HH Intensity factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . . 43

5.34 minor, VH Dispersion factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . 44

5.35 minor, VH Dispersion factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.36 minor, VH Dispersion factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . 44

5.37 minor, HH Dispersion factors (Blue : Bach, Orange : Concone) . . . . . . . . . . . . . 45

5.38 minor, HH Dispersion factors, Bach (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.39 minor, HH Dispersion factors, Concone (Orange) . . . . . . . . . . . . . . . . . . . . . 45

5.40 Major, 0-100 DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.41 Major, Rank DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.42 Major, 0-100 DM and Rank DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.43 minor, 0-100 DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.44 minor, Rank DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.45 minor, 0-100 DM and Rank DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.46 0-100 DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.47 Rank DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.48 The circle of fifths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.49 Bach, Major, VHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.50 Bach, Major, t-DM of VHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.51 Bach, Major, t-DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

LIST OF FIGURES LIST OF FIGURES

5.52 Bach, minor, t-DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.53 Bach, Major, t-DM/1 of VHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.54 Bach, Major, t-DM/t of VHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.55 Bach, Major, t-DM/1 and t-DM/t of VHM (Blue : t-DM/1, Orange : t-DM/t) . . . . 52

5.56 Major, Intensity factors (Blue : t-DM/1, Orange : t-DM/√t) . . . . . . . . . . . . . . 53

5.57 Major, Intensity factors, t-DM/1 (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.58 Major, Intensity factors, t-DM/√t (Orange) . . . . . . . . . . . . . . . . . . . . . . . . 53

5.59 Major, Dispersion factors (Blue : t-DM/1, Orange : t-DM/√t) . . . . . . . . . . . . . 54

5.60 Major, Dispersion factors, t-DM/1 (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.61 Major, Dispersion factors, t-DM/√t (Orange) . . . . . . . . . . . . . . . . . . . . . . . 54

5.62 minor, Intensity factors (Blue : t-DM/1, Orange : t-DM/√t) . . . . . . . . . . . . . . 55

5.63 minor, Intensity factors, t-DM/1 (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.64 minor, Intensity factors, t-DM/√t (Orange) . . . . . . . . . . . . . . . . . . . . . . . . 55

5.65 minor, Dispersion factors (Blue : t-DM/1, Orange : t-DM/√t) . . . . . . . . . . . . . . 56

5.66 minor, Dispersion factors, t-DM/1 (Blue) . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.67 minor, Dispersion factors, t-DM/√t (Orange) . . . . . . . . . . . . . . . . . . . . . . . 56

5.68 Bach, Major, 0-100 Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.69 Bach, minor, 0-100 Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.70 Bach, Major, d(V, V ′) and t-DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.71 Bach, Major, d(V, V ′) and t-DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.72 Bach, Major, d(Vi, V′i ) and t-DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.73 Bach, Major, d(Vd, V′d) and t-DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.74 Bach, minor, d(V, V ′) and t-DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.75 Bach, minor, d(Vi, V′i ) and t-DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.76 Bach, minor, d(Vd, V′d) and t-DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.77 Bach, t-DM of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Chapter 1

Introduction

This chapter includes terminologies in [1] and [2].

1.1 Analysis of musical works

Musical works impress the audience by their own characteristics which can evoke a variety of emotions.

People may explain how they feel with usual words and expressions, whereas experts in music have a

series of terminologies used in their field. This gap between experts and the audience triggered this

study to develop a comprehensible and reasonable method of analysis of musical works.

To be specific, this study aims to analyze musical works based on their score, also called sheet

music, in computational ways with use of computers. Chapter 1 explains temperaments and keys

in music theory, which are essential in this theory, and suggests how to analyze properties of keys.

The foundation of this study from a viewpoint of music and physics is in Chapter 2, followed by

mathematical modeling of analysis in Chapter 3. Then the use of computers is designed in Chapter

4. The analysis with the result of computations will be discussed in Chapter 5.

1.2 Semitones and octaves

In music, a note is the most basic element with two parts, pitch and duration. Pitch means how high

the note is, and duration is how long the note lasts. To understand and explain notes is the most

important base for this study, several pitch-related terminologies are explained below.

1.2.1 Octaves and semitones

Let’s consider the set Z of the integers as an additive group, and the quotient group Z/12. One can

suggest the concept of a period 12 since n and n + 12 are identical in Z/12. With this concept, a

period is formed by 12 steps, from n to n+ 12.

1

1.3. TEMPERAMENTS CHAPTER 1. INTRODUCTION

A very similar system is used in the existing theory of music. The set of pitches is designed to

be discrete with a period called an octave. Within an octave, 12 semitones are used to move from

one pitch to the pitch which is one octave higher than the old one1. These concepts will be explained

more explicitly in the coming topic.

1.2.2 Scientific pitch notation

Let’s go back to Z and Z/12 and consider how to name integers. One canonical way is to borrow

quotient and remainder. Define f : Z → (Z/12) × Z by f(n) = (r, q), where n = r + 12q with q ∈ Z

and r ∈ Z/12 = {0, 1, 2, · · · , 10, 11}. For example, f(12) = (0, 1) and f(23) = (11, 1).

It is a good way to develop a similar system to name pitches with their semitones and octaves.

This system is called Scientific Pitch Notation (SPN), which names pitches like C4, G3, E♭5, A＃4

and so on. Note that the terminology octave is used for not only the period with 12 semitones but

the quotient in the SPN system. The following is the rule of SPN system.

First, consider a set of 12 pitches without octaves, Pitches = { C, C＃/D♭, D, D＃/E♭, E, F,

F＃/G♭, G, G＃/A♭, A, A＃/B♭, B }. Note that expressions like C＃/D♭ show that one pitch can have

different names. Second, define a bijective function f : Pitches ↔ Semitones = {0, 1, · · · , 10, 11}with C ↔ 0,C＃/D♭↔ 1,D ↔ 2, · · · ,A＃/B♭↔ 10,B ↔ 11. Third, as it has been already suggested,

accidentals such as sharp (＃) and flat (♭) can increase or decrease the value of f . To be precise, sharp

(＃) increases the value of f by 1, while flat (♭) decreases. Fourth, combine it with octaves so that

the final result seems like C4, G3, E♭5, A＃4 and so on.

The SPN system has been described above, but one thing remains undecided. When it comes to

octave, a standard is required to say which C is C4 among other C pitches, or which A is A4 among

other A pitches. It is better to move to the next topic in order to explain this matter later.

1.3 Temperaments

1.3.1 Definition of temperaments

In case a musical instrument such as a guitar has been used for a long time, owners should tune it

to make its sound ”correct.” In other words, the process of tuning is adjusting pitches to the correct

ones. Pitch can be expressed in terms of frequency. In other words, tuning a musical instrument for

correct sound is indeed adjusting pitches to the frequencies which have been already fixed or shared

as a standard.

Temperament is a standard of tuning with the list of frequencies for pitches. Every frequecy for

necessary pitches has to belong to a temperament, so that a temperament is a fuction from pitches

1 In fact, an octave doesn’t have to be divided into 12 semitones. However, 12-semitone octave has been an interna-tional standard for hundreds of years. All musical works in this study also employ the same system.

2

1.3. TEMPERAMENTS CHAPTER 1. INTRODUCTION

to frequencies. As a result, a name of a pitch cannot determine its correct frequency, unless the

temperament is specified.

Before specification of temperament, there is one axiom for frequencies. This axiom regulates how

to specify any temperament in terms of frequencies.

Axiom 1.3.1.1 (Octave-Frequency Axiom). The frequency ratio of two pitches is exactly 2 when one

pitch is one octave higher than the other.

Under this axiom, the frequency ratio of C4 over C3 is 2, and it is the same for G4 over G3.

This principle is widely used as a standard, and every temperament in this study obeys it. From this

axiom, the following proposition is clear.

Proposition 1.3.1.1 (Temperament Condition). A temperament is determined by two factors.

1. All frequency ratios of arbitrary two pitches in Pitches = { C, C＃/D♭, D, D＃/E♭, E, F,

F＃/G♭, G, G＃/A♭, A, A＃/B♭, B }2. Frequency of one of pitches in Pitches in any octave

Proof. Consider Z/12 and Semitones = {0, 1, · · · , 10, 11} as a bijective objective of Pitches. Given

two factors in the proposition, we can assume, WLOG, that the frequency of C3 is given and therefore

all frequencies of 12 pitches from C3 to B3 are known. Therefore, the frequency of any pitch in any

octave can be computed by the use of Octave-Frequency Axiom.

Proposition 1.3.1.2. The frequency ratio of two pitches does not change when the pitches are shifted

to another octave.

This proposition means that, for example, the frequency ratio of G3 over C3 is equal to that of

G4 over C4. It is also directly proved by Octave-Frequency Axiom.

By Temperament Condition, when at least one frequency of one pitch is given, the study of

temperaments can only focus on the frequencies within an octave. In fact, one standard frequency of

one pitch now exists for A4 = 440 Hz. Furthermore, this study will not be affected by the change of

one standard frequency. in terms of computations this study will suggest. Consequently, the remaining

part of this section 1.3 can focus on the frequencies within an octave.

1.3.2 Equal temperament & pure intonation

Equal temperament is currently a standard temperament. As its name indicates, the frequencies in an

octave are distributed with the equal ”distance.” More precisely, the frequency ratio for a semitone

is constant between any two consecutive pitches.

A direct result is that the frequency ratio for a semitone is 2112 , so that an octave has the frequency

ratio 2 = (2112 )12. With this value for a semitone, one can compute the frequency of any pitch from

that of A4 = 440 Hz, such as B4 = 440 · (2 112 )2 Hz where B4 is two semitones higher than A4. Note

3

1.4. KEYS AND THE CIRCLE OF FIFTHS CHAPTER 1. INTRODUCTION

that (2112 )k is irrational for k = 1, 2, · · · , 10, 11. In other words, the frequency ratio of two pitches,

say P1 and P2, is rational iff P1 is n octaves higher or lower than P2 for some nonnegative integer n.

On the contrary, pure intonation is a temperament where every frequecy ratio is rational, being

the opposite concept of equal temperament. While equal temperament is fixed as one system, pure

intonation can have a variety by the choice of rational frequency ratios. Instead of selecting one pure

intonation system, it is important to remember, in this study, that the most important concept of pure

intonation is the rational frequency ratio between any two pitches. Note that these two temperament

systems are not totally different, and this will be explined in section 2.3

1.4 Keys and the circle of fifths

In the previous section, the concept of temperament was explained. The usage of temperaments has

to wait for a moment until section 1.5 though. This section and the next section are going to cover

keys and their properties, which are the main target of this study.

1.4.1 Definition of keys

Key is a terminology for a theoretical concept in music. In terms of pitches, a key is a set of pitches

that will form a musical work or a part of it. As mentioned at the beginning of section 1.2, notes and

their pitches are basic elements for musical works. Therefore, a key provides a framework of pitches

in which a musical work can choose notes. Remember that a key is a theoretical concept because it

cannot make actual sounds unless a temperament assigns frequencies to pitches.

When a temperament is explained in the previous section, Temperament Condition says that it is

enough to determine frequencies within an octave. A similar principle is valid for keys in this study,

and it is suggested as a remark.

Remark 1.4.1.1 (Key Condition). A key is determined by the choice of pitches from Pitches = { C,

C＃/D♭, D, D＃/E♭, E, F, F＃/G♭, G, G＃/A♭, A, A＃/B♭, B }This principle is widely used as a standard, and all keys in this study follows it. As a result, the

only thing to determine a key is to choose necessary pitches within an octave.

1.4.2 Major and minor

Two commonly used types of keys are major and minor. Both types consist of 7 pitches within an

octave, and their choice is different. To explain the difference, it is useful to express major and minor

with the bijection Pitches ↔ Semitones = {0, 1, · · · , 10, 11}It is better to introduce two terminologies to explain major and minor more clearly. An interval

means the distance between two pitches. A semitone is obviously one of intervals used in music theory.

A degree is an order of positions of pitches in a key. To have 7 pitches in major or minor means to

4


have 7 notes in 1st, 2nd, 3rd, 4th, 5th, 6th, and 7th degree. Now let’s explain major and minor in

terms of pitches, degrees, and intervals.

• Major consists of 7 pitches from 1st degree to 7th degree (8th degree for the next octave)

0, 2, 4, 5, 7, 9, 11, (12 for 8th degree) in Semitones, with the sequence of 7 intervals (from 1st

degree to 8th degree : 7 intervals among 8 pitches) 2, 2, 1, 2, 2, 2, 1 in semitones.

• If major starts with C, then it consists of C, D, E, F, G, A, B, (C for the next octave). In terms

of intervals, 5 intervals C-D, D-E, F-G, G-A, A-B are in 2 semitones, while 2 intervals E-F, B-C

are in 1 semitone.

• Minor consists of 7 pitches from 1st degree to 7th degree (8th degree for the next octave)

0, 2, 3, 5, 7, 8, 10, (12 for 8th degree) in Semitones, with the sequence of 7 intervals (from 1st

degree to 8th degree : 7 intervals among 8 pitches) 2, 1, 2, 2, 1, 2, 2 in semitones.

• If minor starts with A, then it consists of A, B, C, D, E, F, G, (A for the next octave). In terms

of intervals, 5 intervals A-B, C-D, D-E, F-G, G-A are in 2 semitones, while 2 intervals B-C, E-F

are in 1 semitone.

It seems that C major and A minor consist of the same pitches, except their degrees. However,

minor keys are more complex with their variations. Minor keys have three typical variations, and these

variations are used together in musical works. In other words, minor key is a mixed key of variations,

not one fixed key. Here is a short description of variations of minor.

• Natural minor has been already introduced above as minor, so that it is just renamed. It is also

the foundation of the following variations

• Harmonic minor is obtained from natural minor when 7th degree pitch becomes higher by one

semitone. If natural minor consists of A, B, C, D, E, F, G, (A for the next octave), then harmonic

minor consists of A, B, C, D, E, F, G＃, (A for the next octave), with the sequence of 7 intervals

2, 1, 2, 2, 1, 3, 1 in semitones.

• Melodic minor is divided into two cases, and it is decided by wheter the melody is ascending or

descending.

• When the melody is ascending, it is obtained from harmonic minor when 6th degree pitch

becomes higher by one semitone. If harmonic minor consists of A, B, C, D, E, F, G＃, (A for the

next octave), then melodic minor consists of A, B, C, D, E, F＃, G＃, (A for the next octave),

with the sequence of 7 intervals 2, 1, 2, 2, 2, 2, 1 in semitones.

• When the melody is descending, it is just the same as natural minor.

To summarize, minor key is a mixed key of its variations and is different from major key. One

more thing is how to name major and minor keys based on their 1st degree pitches. When major key

starts with C as it 1st degree pitch, then it is called C major. Minor key follows the same rule to have

A as its 1st degree pitch for the case of A minor. Moreover, every key has its 1st degree pitch and it

has a special name tonic. It means that the name of major or minor key is determined by its tonic.

5


There are still many things to explain about major and minor keys, such as how C major and G

major are similar or different, or how C major and A minor are connected. The next topic will cover

these questions.

1.4.3 Circle of fifths

Let’s start with two remarks.

Remark 1.4.3.1. Two major keys with different tonics have the same sequence of 7 intervals in semi-

tones within an octave from its tonic to the pitch one octave above.

Remark 1.4.3.2. Two minor keys with different tonics have the same sequence of 7 intervals in semi-

tones within an octave from its tonic to the pitch one octave above.

For example, C major consists of C, D, E, F, G, A, B, C with the sequence of 7 intervals 2, 2, 1,

2, 2, 2, 1 in semitones. Similarly, G major consists of G, A, B, C, D, E, F＃, G with the same sequence

of 7 intervals in semitones. The same principle also works for minor keys, including the variations. It

is useful to rewrite the result of remarks in the form of propositions.

Proposition 1.4.3.1 (Major Shift Principle). A major key can be shifted to any major key by the

shift of tonic.

Proposition 1.4.3.2 (Minor Shift Principle). A minor key can be shifted to any minor key by the

shift of tonic.

As a result, 12 major keys are connected to each other by shifts of tonics, and so are 12 minor keys.

Once again, these shift principles are in theoretical sense of pitches, not related with temperaments.

In section 1.5, the connection between keys and temperaments will be dicussed.

Going back to major and minor keys, let’s take a look at the connection between major and minor.

When it comes to minor keys, let’s be based on natural minor keys. For example, C major and A

minor share the same set of pitches {A, B, C, D, E, F, G}, although degrees of pitches are different.

Another example is G major and E minor with the same set of pitches {E, F＃, G, A, B, C, D}.The examples here show a pair of major and minor keys with the same set of pitches. In other

words, 12 major keys and 12 minor keys can be classified into 12 pairs of major and minor keys so

that for a fixed pair, major and minor keys share the same key signature. A key signature is a set of

sharp (＃) and flat (♭) in which a key can find their pitches. In the case of C major and A minor, no

sharp or flat is necessary, while one sharp is required for G major and E minor.

Let’s introduce some terminologies about these pairs. A major key is the relative major of a minor

key if two share the same key signature. The relative minor is defined in the same manner. With the

concept of relative major/minor, there is a way to visualize pairs of major/minor, called the circle of

fifths. Let’s take a loot at a figure2 and discuss more.

2https://upload.wikimedia.org/wikipedia/commons/3/33/Circle_of_fifths_deluxe_4.svg

6


Figure 1.1: The circle of fifths

Major keys are outside the circle, while minor keys are insie the circle. Starting from C major /

A minor, go around the circle clockwise. Increasing the number of sharps by 1, the situation gets a

little confusing by showing both 5 sharps and 7 flats together for B major (C♭ major) / G＃ minor. It

occurs because for this pair of major/minor keys, two key sigatures are possible. The similar situation

runs for three pairs of major/minor.

B major (C♭ major) / G＃ minor : 5 sharps or 7 flats

G♭ major (F＃ major) / E♭ minor (D＃ minor) : 6 sharps or 6 flats

D♭ major (C＃ major) / B♭ minor : 7 sharps or 5 flats

7

1.5. PROPERTIES OF KEYS CHAPTER 1. INTRODUCTION

During these pairs, the number of sharps is increasing by 1, while the number of flats is decreasing

by 1. After them, the number of flats decreases by 1 until it arrives at C major / A minor where

no sharp or flat exists. Now Major Shift Principle and Minor Shift Principle can be visualized in

the circle of fifths with the information of how many sharps or flats should be added or deleted. In

addition, this circle of fifths can suggest a idea of ”close” keys and ”distant” keys. This idea will be

explained in detail in the next section.

1.5 Properties of keys

To introduce one additional terminology, a shift from a key to another key is called modulation.

Major Shift Principle, Minor Shift Principle, and the circle of fifths show how modulation can be

done. However, recall what has been explained in section 1.3. Equal temperament and pure intonation

assign different frequencies to pitches, inducing modulation to be reconsidered.

In history, pure intonation started to be used earlier than equal temperament. An advantage of

pure intonation is that it is easy to tune pitches because the frequency ratios are rational. At the

same time, the frequency ratios of semitones have to be different, making modulation limited in

the real world. As equal temperament appeared, this system realized free modulation among keys.

Theoretical concept of keys and the circle of fifths could be a practical manner in which musicians

can do modulation.

Remember this change in modulation, because the remaining part of this section will suggest

properties of keys which are connected to the difference between pure intonation and equal temper-

ament. Three properties of keys are suggested and they will be analyzed in this study by modeling

and computer process.

Major and minor

Major keys and minor keys are different even in theoretical concept of keys and pitches regardless

of temperaments. This study will show the difference of major and minor keys in several factors.

Key characteristic

From pure intonation and different frequency ratios of semitones, different major keys (minor

keys) had different characteristics. On the other hand, equal temperament makes all major keys

(minor keys) equal except their different tonics. This study will check whether key characteristic

does exist.

Key distance

The circle of fifths visualizes 12 major keys and 12 minor keys according to the change of key

signature. In addition, among all major keys (minor keys), there are ”close” keys and ”distant”

keys based of the circle of fifths in music theory of modulation. This study will see whether the

concept of key distance in the circle of fifths is realistic.

8

1.5. PROPERTIES OF KEYS CHAPTER 1. INTRODUCTION

Three properties of keys are the main targets of this study. In order to analyze them, let’s introduce

briefly how to analyze.

• Keys are concepts in music theory, not musical works. That’s why this study needs musical

works which can be good materials to analyze. For 24 keys with 12 major keys and 12 minor

keys, there are sets of 24 musical works written in all major and minor keys.

• Two sets by two different composers are used in this study. The primary set is Preludes in

Well-Tempered Clavier, BWV 846-869, by Johann Sebastian Bach (1685-1750)3. The

secondary set is Preludes, Op. 37, by Giuseppe Concone (1801-1861)4.

• The set of Bach will be used for three properties, while the set of Concone is used only for Key

characteristic.

• While analyzing musical works, the real target of this study is keys, not musical works. This

philosophy is very important and it will determine the foundation of this study in the next

chapter, especially in section 2.3.

• Although musical works consist of many components including pitch, harmony, rhythm, dynam-

ics, tempo, timbre and others, this study only computes pitch and harmony. Once again, it is

from the philosophy that the real target is keys, not musical works.

Let’s conclude this chapter by writing two remarks.

Remark 1.5.0.1 (Key Philosophy). The real target of this study is keys, not musical works.

Remark 1.5.0.2 (Pitch Harmony Philosophy). This study computes pitch and harmony.

3http://ks.imslp.info/files/imglnks/usimg/9/9b/IMSLP411479-PMLP05948-bach-wtk-ur-1.pdf4http://ks.imslp.net/files/imglnks/usimg/e/e2/IMSLP09618-Concone_-_Op.37_-_24_Preludes.pdf

9

Chapter 2

Foundation

Remember Key Philosophy and Pitch Harmony Philosophy. Although this study is for computational

analysis, it’s a good idea to understand how classical theory of music has analyzed keys and musical

works. This chapter will develop a method to analyze intervals in a computational way, with a

viewpoint of music and physics.

2.1 Classical theory of consonance and dissonance

Recall that an interval is the distance between two pitches. Pitches and intervals are the elements

to form harmony, which is essential to analyze keys. Classical theory of music has the concept of

consonance and dissonance. By Key Philosophy, intervals within an octave should be classified. To be

precise, it is enough to classify intervals of s semitones with s = 0, 1, 2, · · · , 10, 11, 12. Intervals are

classified into consonance and dissonance, and consonance has two subclassification. The following is

classification into three categories.

Perfect consonance

s = 0, 5, 7, 12

Imperfect consonance

s = 3, 4, 8, 9

Dissonance

s = 1, 2, 6, 10, 11

This is a common classification used in classical theory of music and harmonics, from page 9 of

[3]. These three categories will be a standard throughout this chapter, especially in section 2.3 for the

design of computation of harmony. Call the classification Consonance Dissonance Classification.

10

2.2. WAVE AND FREQUENCY RATIO CHAPTER 2. FOUNDATION

2.2 Wave and frequency ratio

2.2.1 Wave and superposition

Imagine that two sound waves are coming. One can write them as W1(t) = A1 sin(2πf1t + c1) and

W2(t) = A2 sin(2πf2t + c2) in a simple form. By Key Philosophy, it is good to express sound waves

in that simple form. Once again, recall Pitch Harmony Philosophy. Now assume A1 = A2 = 1 and

c1 = c2 = 0 for convenience and compute the following.

W1(t) +W2(t) = sin(2πf1t) + sin(2πf2t)

= 2 sin2πf1t+ 2πf2t

2cos

2πf1t− 2πf2t

2

= 2 sin{(f1 + f2)2πt

2} cos{(f1 − f2)

2πt

2} (2.1)

2.2.2 Frequency ratio

In terms of frequency ratio, the choice of temperaments is really important. When equal temperament

is chosen, the frequency ratios within an octave will be irrational except two pitches at the end of an

octave. On the contrary, if pure intonation is selected, every frequency ratio of an interval within an

octave should be expressed as a rational number.

By the way, the most important criterion to choose temperaments should be the goal of analysis.

Remember that this study is focused on three properties of keys suggested in section 1.5, which

are Major and minor, Key characteristic, and Key distance. Among three properties, Key

characteristic is directly influenced by the choice of temperaments. Equal temperament is designed

to show no differences between all major keys (minor keys), while pure intonation should make

different characteristics for keys from its uneven semitones in terms of frequency ratios. The other

properties mainly depend on theoretical pitches and keys.

In order to verify Key characteristic, this study will choose pure intonation. If the result of this

study shows no meaningful characteristic even in pure intonation, then the conclusion will be to say

that Key characteristic is not realistic. On the contrary, if the result shows reasonable characteristic

in pure intonation, then Key characteristic is justified at least for pure intonation, but nothing is

guaranteed for equal intonation.

Let’s move to the next topic after a simple computation. For two frequencies f1 and f2, assume

that f1 : f2 = n : m with n,m ∈ N and n < m. Write f1 = nl and f2 = ml for a positive l ∈ R. Then

the formula 2.1 can be express as follows.

W1(t) +W2(t) = 2 sin{(n+m)2πl

2t} cos{(n−m)

2πl

2t} (2.2)

11

2.3. INTERVAL HARMONY (IH) CHAPTER 2. FOUNDATION

2.3 Interval Harmony (IH)

2.3.1 Interval Distance (ID)

Given two pitches, the interval can be expressed in semitones. For example, the interval between C4

and G4 is 7 semitones, and the interval between C4 and G5 is 19 semitones. This is a direct way

to indicate the interval from a viewpoint of physics. Once again, remember Key Philosophy. It is

reasonable to consider G4 and G5 equivalent to each other, as it was used in [4]. Following this idea,

interval distance (ID) between two pitches, say P1 and P2, is defined as follows.

Definition 2.3.1.1 (Interval Distance (ID)). Let L be the interval between P1 and P2 in semitones.

ID(P1, P2) = ID(L) =

⎧⎨⎩0 if L = 0

R(L) if L > 0

where, for L > 0, L = 12Q(L) +R(L)

with Q(L), R(L) ∈ Z, 1 ≤ R(L) ≤ 12

For example, ID = 0 for L = 0, ID = 12 for L = 12n for n ∈ N, ID = 7 for L = 7, ID = 7 for L

= 12n + 7 for n ∈ N. In other words, when L > 12, decrease L by 12 repeatedly until L ≤ 12. This

definition of ID is designed to shift intervals inside an octave with 12 semitones. More examples with

pitches are here. ID(C4, C4) = 0, ID(C4, C5) = ID(C4, C6) = 12, ID(C4, G4) = ID(C4, G5) = 7.

2.3.2 Frequency ratios

From the definition of ID, it is now necessary to convert ID to something new which can provides

meaningful computation of harmony. First of all, recall the formula 2.2 again, and note that |n+m| >|n−m| and 2πl

2 t is common for sin and cos function.

W1(t) +W2(t) = 2 sin{(n+m)2πl

2t} cos{(n−m)

2πl

2t} (2.3)

Since classical theory of music and harmonics use words tension and resolution to explain how

musical works use consonance and dissonance, as shown in page 110 of [1], this study needs a way to

compute tension of harmony. This is where a viewpoint of physics is necessary.

When there is a sound, it delivers energy. The intensity of sounds is power per unit area, that is, the

amount of energy delivered per unit time and unit area, explained at page 99 of [5]. Note intensity =energy

time·area is proportional to the kinetic energy 12mv2, where v = d

dt sin(2πft) = 2πf sin(2πft) is

proportional to the value of the frequency. Consequently, the intensity is proportional to the square of the frequency. By Key Philosophy, this study is interested in only frequencies. It is better to write it as a remark for this study.

Remark 2.3.2.1 (Intensity Frequency Principle). The intensity of a sound is proportional to the square

of the frequency.

12


Now take a look at the formula 2.3 again. Since |n+m| > |n−m|, by Intensity Frequency Principle,

one can say that the term sin{(n +m) l2 t} is dominant in this formula. Let’s focus on the frequency

(n+m) l2 of this sin term.

Remember that n and m are natural numbers to indicate n : m = f1 : f2 with f1 = nl and

f2 = ml. The value of l varies according to the choice of octaves. For example, if two pitches P1 and

P2 have their frequencies f1 = nl and f2 = ml, then the pitches P1- and P2-, which are one octave

lower than P1 and P2 respectively, have the frequencies f1− = nl2 and f2− = ml

2 . Considering this fact

with Key Philosophy, it is good to choose n and m for the frequency ratio, with n and m relatively

prime, so that (n,m) can be applied to pitches regardless of octaves. This manner is compatible

with the range of ID = {0, 1, 2, · · · , 10, 11, 12} within an octave. From now on, n and m are always

relatively prime.

2.3.3 Interval Harmony (IH)

Now the important task is to assign (n,m) to every value of ID. As mentioned before, one should

focus on the dominant term sin{(n+m) l2 t} in the formula 2.3, and in addition, just take a look at the

(n+m) part instead of the frequency (n+m) l2 , so that (n+m) can provide a criterion in choosing

(n,m). For the convenience, denote k = n +m. In other words, the coming task is to assign (n,m)

to the range of ID, with k being a standard for a frequency. Before the task starts, two things are

needed to be called again.

First, recall Consonance Dissonance Classification in section 2.1, where s is the interval in semi-

tones within an octave.

Perfect consonance

s = 0, 5, 7, 12


s = 3, 4, 8, 9

Dissonance

s = 1, 2, 6, 10, 11

Second, Intesity Frequency Principle is restated here with k.

Remark 2.3.3.1 (Intensity Frequency Principle). The intensity of a sound is proportional to k2.

Define interval harmony as IH = k2. Now the main task is to choose (n,m) with IH for s in

{0, 1, 2, · · · , 10, 11, 12}. Then two questions arise as follows.

1. As mentioned in section 2.2, choosing n,m in N is for pure intonation. Is it totally independent

of equal temperament? This question is also connected with section 1.3.

2. Consonance Dissonance Classification should be considered when choosing (n,m) and IH. How

to connect IH with Consonance Dissonance Classification?

13


Let’s describe the procedure of the main task. The procedure will naturaly answer two questions.

First of all, let’s take a look at equal temperament within an octave. As explained in section 1.3, every

semitone has the same frequency ratio 2112 . Denote r = 2

112 and write an octave within 12 semitones in

relative frequencies starting with 1 as {1 = r0, r1, r2, · · · , r10, r11, r12 = 2}. For convenience, converta frequency f , to 1200 log2 f , and the converted result is {0, 100, 200, · · · , 1000, 1100, 1200}. It is easyto use because one semitone in equal temperament can be seen as 100, because rs is converted to

1200 log2 rs = 1200 log2 2

s12 = 1200 · s

12 = 100s.

Now consider (n,m) as f1 : f2 = n : m in the pure intonation. From now on, WLOG, assume

n ≤ m ≤ 2n because the computation is within an octave. The frequency ratio f2f1

= mn is now

converted to 1200 log2mn . It is better to summarize these computations as follows.

1. In equal temperament, 12 semitones are converted to {0, 100, 200, · · · , 1000, 1100, 1200}2. For f1 = 1 and f2, f2 is converted to 1200 log2 f2

3. In the same setting, if f1 : f2 = n : m, then f2 is converted to 1200 log2mn

Call this conversion from frequency octave to linear octave. Once again, remember that this conversion

is relative within an octave, having f1 = 1 as the origin with 1200 log2 f1 = 0.

With this conversion from frequency octave to linear octave, the main goal is to convert intervals

of s semitones for s in {0, 1, 2, · · · , 10, 11, 12} using (n,m) for pure intonation. There are 13 intervals

but two cases for s = 0, 12 are trivial. The interval s = 0 is converted to 0 with (1, 1), and the interval

s = 12 is converted to 1200 with (1, 2). The other intervals s = 1, 2, · · · , 10, 11 are the targets.

At this point, the method to find (n,m) for intervals depends on equal temperament and Conso-

nance Dissonance Classification. Since equal temperament was developed to replace or modify pure

intonation1, two systems share the same 12 semitones within an octave, with some distances in their

frequencies. Therefore, this study will find (n,m) based on equal temperament, by setting reasonable

limit of frequency errors. Here is a list of conditions of optimization process.

1. Find (n,m) pairs with n,m relatively prime and n < m < 2n. The cases m = n, 2n are for

s = 0, 12, which are not targets.

2. By conversion from frequency octave to linear octave, classify pairs according to the nearest

frequencies in equal temperament.

3. The boundary of k (BoK) and the limit of frequency errors (LoFE) need to be decided.

4. LoFE must be less than 1003 in linear octave.

5. BoK should be decided so that every interval s = 1, 2, · · · , 10, 11 have at leats one (n,m) pair.

6. Each interval chooses the smallest k within BoK and LoFE.

7. The collection of k must satisfy Consonance Dissonance Classification.

8. Imperfect consonance and dissonance have to be distinguished explicitly in terms of IH.

1The history of pure intonation is very long, at least the same as Pythagoras.

14


Condition 4 comes from this reason. Since LoFE is less than 1003 , when two pitches P1 and P2

have their frequencies for 100 and 200 in linear octave, P1 < 100 + 1003 and P2 > 200 − 100

3 so that

P2 cannot be closer than 100 for P1, and P1 cannot be closer than 200 for P2.

Now let’s see how this procedure has gone. The first step is to find BoK satisfying Condition 5.

The value is BoK = 25, and the result is here.

Figure 2.1: BoK=25, LoFE=33

It has been done in Microsoft Excel. Within BoK = 25, all pairs of (n,m) and the result of

conversion from frequency octave to linear octave are classified according to the nearest intervals

among s = 1, 2, · · · , 10, 11, with errors written together. For each interval, one pair of (n,m) is selected

under Condition 6, being in bold. The result is classified into three groups, Perfect consonance

(upper left), Imperfect consonance (lower left), and Dissonance (right).

Two problems exist in this setting. First, the interval s = 1 (100 in linear octave) has only one

pair of (n,m) and its error exceeds LoFE. Second, s = 6 has selected the pair (5, 7) with k = 12,

which is smaller than k = 13 of (5, 8) in s = 8, violating Condition 7.

To solve the second problem violating Condition 7, just increasing BoK isn’t a solution, because

it can’t change the choice of (n,m) pairs for intervals which have already picked their pairs within

15


LoFE. In this case, decreasing LoFE cannot be a solution due to the interval s = 1. As a result, the

only solution is to increase BoK and decrease LoFE at the same time.

Let’s figure out the required change of BoK and LoFE exactly. The interval s = 6 has k = 12 and

an error greater than 17.4, which implies that LoFE should be less than 17.4, forcing the interval s = 6

to pick (n,m) with a larger k. At the same time, LoFE less than 17.4 requires bigger BoK so that

the interval s = 1 can select (n,m) with k less than 17.4 within BoK. Combining these requirements,

the minimum value of BoK is 31. Let’s see the case.

Figure 2.2: BoK=31

16


Now let’s try to find a suitable value of LoFE in BOK = 31. The given condition is LoFE < 17.4

and turn attention to the interval s = 9. If LoFE is less than the error in s = 9, then s = 9 has no

(n,m) pairs to choose in BoK = 31. Assume 15.7 < LoFE < 17.4 and observe that there is no (n,m)

pairs with k in the range of LoFE. To simplify the value, let LoFE = 16 and see what happens.

Figure 2.3: BoK=31, LoFE=16

Every interval can find (n,m) pairs within LoFE. Now check Condition 7 and Condition 8. Re-

member that the result is classified into three groups, Perfect consonance (upper left), Imperfect

consonance (lower left), and Dissonance (right). The values of k and IH are listed below according

to groups, including s = 0 with (1, 1) and s = 12 with (1, 2) together.

17


Perfect consonance

s = 0, 5, 7, 12

k = 2, 7, 5, 3 with minimum = 2, maximum = 7

IH = 4, 49, 25, 9 with minimum = 4, maximum = 49


s = 3, 4, 8, 9

k = 11, 9, 13, 8 with minimum = 8, maximum = 13

IH = 121, 81, 169, 64, with minimum = 64, maximum = 169

Dissonance

s = 1, 2, 6, 10, 11

k = 31, 17, 29, 25, 23 with minimum = 17, maximum = 31

IH = 961, 289, 841, 625, 529 with minimum = 289, maximum = 961

From the values of minima and maxima of k and IH, Condition 7 and Condition 8 are satisfied.

Consequently, the result can be used as the values of IH from the optimization process satisfying

Condition 1 to Condition 8. Conclude this section with the function from ID to IH as follows.

Definition 2.3.3.1 (ID to IH (Interval harmony)).

R(ID) = {0, 1, 2, · · · , 10, 11, 12}F : R(ID) → N

F (0) = 4, F (1) = 961, F (2) = 289, F (3) = 121, F (4) = 81,

F (5) = 49,F (6) = 841, F (7) = 25, F (8) = 169,

F (9) = 64, F (10) = 625, F (11) = 529, F (12) = 9

18

Chapter 3

Modeling

In the previous chapter, every interval between two pitches was converted to ID within an octave,

and the function F from ID to IH was also defined within an octave. Every modeling in this study

will be based on ID and IH. Remember Pitch Harmony Philosophy, so that the modeling is about

pitches and harmony.

3.1 Pitches to Pitch Value (PV)

The first step is to convert the name of pitch to the number of semitones. This conversion is necessary

because an interval between two pitches needs to be expressed in semitones so that ID and IH can be

computed. Pitch value (PV) is now defined as follows.

Definition 3.1.0.1 (Pitch value (PV)). Write the name of a pitch P in the SPN system with two

components, NoP (note of pitch) and OoP (octave of pitch). For C4, C is NoP and 4 is OoP.

1. PV(P) = PV(NoP) + 12 · OoP

2. PV(NoP) is defined as follows.

PV(C) = 0, PV(D) = 2, PV(E) = 4, PV(F) = 5, PV(G) = 7, PV(A) = 9, PV(B) = 11

3. PV(NoP-sharp) = PV(NoP) + 1, PV(NoP-flat) = PV(NoP) - 1

4. PV(NoP-doublesharp) = PV(NoP) + 2, PV(NoP-doubleflat) = PV(NoP) - 2

Let’s see some examples.

PV(C4) = 0 + 12 · 4 = 48, PV(G4) = 7 + 12 · 4 = 55, PV(B4) = 11 + 12 · 4 = 59

PV(C♭4) = 1 + 12 · 4 = 47, PV(C＃4) = 1 + 12 · 4 = 49, PV(B＃4) = 12 + 12 · 4 = 60

PV(C♭♭4) = -2 + 12 · 4 = 46, PV(C＃＃4)1 = 2 + 12 · 4 = 50, PV(B＃＃4) = 13 + 12 · 4 = 61

Note that PV is well-defined for the name of pitches, that is, when a pitch has two different names,

being enharmonic, the value of PV doesn’t change.

1 There is a notation for doublesharp instead of ＃＃, but not used here.

19

3.2. PV TO ID TO IH CHAPTER 3. MODELING

The role of PV is to express pitches in the number of semitones. For the use of PV in this study,

the value itself is not used, but the difference is important. This is because intervals are the elements

to compute harmony. More explanation will be written in the next section.

3.2 PV to ID to IH

This section will use PV to compute ID and IH of intervals. Say two pitches P1 and P2 are given and

the values of PV(P1) and PV(P2) can be computed by definition in the previous section. Since the

interval between P1 and P2 can be expressed in semitions by the difference of PV(P1) and PV(P2),

ID(P1, P2) = ID(|PV (P1)−PV (P2)|) can be computed by 2.3.1.1 in section 2.3. Once ID(P1, P2)

is computed, one is able to get IH = F (ID(P1, P2)) = F (ID(|PV (P1) − PV (P2)|)) by 2.3.3.1 in

section 2.3 again. For the convenience of notation, write IH of P1 and P2 as follows.

Notation 3.2.0.1 (IH of P1 and P2). IH(P1, P2) = F (ID(|PV (P1)− PV (P2)|))

This computation of IH(P1, P2) will be a building element for the rest of this chapter. With the

result of 2.3.1.1 and 2.3.3.1 in section 2.3, compute IH with the examples of 3.1.0.1 in section 3.1.

IH(C4, C4) = F(ID(|48− 48|)) = F(ID(0)) = F(0) = 4 = IH(G4, G4)

IH(C4, C5) = F(ID(|48− 60|)) = F(ID(12)) = F(12) = 9 = IH(C4, C6) = IH(G4, G5)

IH(C4, G4) = F(ID(|48− 55|)) = F(ID(7)) = F(7) = 25 = IH(C4, G5)

Proposition 3.2.0.1. Here are some properties of IH.

1. IH(P1, P2) = IH(P2, P1) (Symmetric)

2. IH(P1, P2) = 4 ⇐⇒ PV(P1) = PV(P2) (Identical PV)

3. IH(P1, P2) = 9 ⇐⇒ |PV (P1)− PV (P2)| = 12l for some l ∈ N (Octave-distant)

Proof. Obvious from the properties of ID.

3.3 Vertical Harmony (VH) and Horizontal Harmony (HH)

In the previous section, the method to compute IH of two pitches was summarized. Recalling Pitch

Harmony Philosophy, IH is not enough to analyze harmony, because more than two pitches can sound

together and the combination of pitches varies. This section and the next section aim to design more

methods to analyze harmony using IH.

Musical works consist of pitches in time. Pitches appear, last, and disappear as time goes on.

As a result, harmony in musical works can have two dimensions, vertical and horizontal. Horizontal

dimension is for ongoing time, while vertical dimension is for a moment with fixed time. As a result,

vertical harmony (VH) is interested in a combination of pitches which sound together at a moment,

whereas horizontal harmony (HH) observes how combinations of pitches are changing.

20

3.4. MEAN AND SUM OF VH AND HH CHAPTER 3. MODELING

3.4 Mean and sum of VH and HH

With the concept of VH and HH from the previous section, let’s see how to build computational

methods for VH and HH using IH.

3.4.1 VH

One may consider t notes sounding together at a moment. Choosing a pair of two distinct notes in t

notes, the number of all possible pairs is(t2

)= t(t−1)

2 and each pair provides an interval of two notes.

With this situation, define it, the number of intervals, as follows.

it =

⎧⎨⎩

t(t−1)2 if t ≥ 2

1 if t = 1(3.1)

Since t(t−1)2 = 0 for t = 1, set it = 1 for t = 1. Now define two formulas for VH.

Definition 3.4.1.1 (Vertical Harmony Sum (VHS)). VHS =∑

IH

When t > 2, VHS is the sum of IH of t(t−1)2 intervals of two distinct notes. When t = 1, VHS is

equal to 4, the value of Identical PV in proposition 3.2.0.1. The case of t = 1 is defined in this way

as an exception to have nonzero value. With this definition of VHS, its mean version is defined.

Definition 3.4.1.2 (Vertical Harmony Mean (VHM)). VHM = VHS/it

3.4.2 HH

Now consider t1 notes and t2 notes respectively, and assume that the combitation of notes changes

from t1 notes to t2 notes. One can choose a pair of two notes, one in t1 notes and the other in t2 notes.

The number of all possible pairs is t1t2 and each pair has an interval of two notes. In this setting,

define it1t2 , the number of intervals, as follows.

it1t2 = t1t2 (3.2)

Define two formulas for HH, in a similar way as VH.

Definition 3.4.2.1 (Horizontal Harmony Sum (HHS)). HHS =∑

IH

Definition 3.4.2.2 (Horizontal Harmony Mean (HHM)). HHM = HHS/it1t2

HHS is the sum of IH of t1t2 intervals of two notes, one from t1 notes and the other from t2 notes.

Unlike VH, the cases t1 = 1 or t2 = 1 don’t need an exception.

21

3.5. INITIAL RHYTHM SET (IRS) CHAPTER 3. MODELING

3.5 Initial Rhythm Set (IRS)

In the previous section, the concept of VH and HH was explained. To make it clear, this section

introduces one more concept to understand how to compute VH and HH in a musical work.

As mention in section 3.3, notes appear, last, and disappear in a musical work. In order to analyze

VH and HH in this study, every moment when a new note appears will be counted. Initial rhythm set

(IRS) is defined to observe those moments at which VH and HH will be computed.

Figure 3.1: Prelude in BWV 847

Figure 3.1 shows the opening two bars of Prelude in BWV 847. Looking at the first bar, note

that it contains 16 IRS, not 32 IRS although 32 notes appear. This is because 32 note appear in 16

pairs of two notes in order. Therefore, these two bars contain 32 IRS in total, meaning that 32 VH

and 31 HH can be computed. Summarize these properties in the following remark.

Remark 3.5.0.1. [Properties of IRS]

1. When two or more notes appear at the same moment, it deserves exactly one IRS.

2. (# of VH) = (# of IRS)

3. (# of HH) = (# of IRS) - 1

3.6 Dispersion of VH and HH

From section 3.4 and 3.5, a basic structure of VH and HH in a given musical work became practical.

A series of VH and HH can be obtained and computed. Factors such as VH and VH, or more precisely

VHS, VHM, HHS, HHM, will appear from a musical work. One more thing to do with these factors

is to classify them and analyze how dispersed they are. This section will define dispersion for a given

factor of a musical work.

3.6.1 Abstract concept of dispersion

Suppose that datapoints can be distinguished and classified. In addition, assume that different types

of datapoints have a distance. With this setting, one can construct a graph where vertices represent

22

3.6. DISPERSION OF VH AND HH CHAPTER 3. MODELING

types. Moreover, in order to make the graph vertex-edge-weighted, multiplicities are assigned to

vertices and distances are assigned to edges.

Let’s construct the situation more precisely. Suppoese there are m distinguishable datapoints and

they are classified into n types. Type i has its multipliity mi, that is, mi datapoints are classified as

type i. It is clear that m1 +m2 + · · ·+mn = m. Denote the distance between type i and j by d(i, j).

Under this setting, consider a graph with the following properties.

• Vertex vi represents type i, with its vertex weight mi.

• The sum of vertex weights is m = m1 +m2 · · ·+mn

• It is a complete graph with n vertices.

• The edge (vi, vj) has its edge weight d(vi, vj) meaning the distance.

Define a weighted distance d̄{(vi,mi), (vj ,mj)} = d(vi, vj)/w(mi,mj), where w(mi,mj) =mi+mj

m

is the ratio of mi+mj to m. The smaller w(mi,mj) is, the bigger d(vi, vj)/w(mi,mj) becomes. With

this definition of the weighted distance, define the dispersion as follows.

Definition 3.6.1.1. [Dispersion] D =1

n

∑i �=j

d̄{(vi,mi), (vj ,mj)}

In this summation, the number of terms to sum is(n2

)= n(n−1)

2 , being a quadratic polynomial in

n. That’s why the summation is divided by n, making D a linear function in n. Combining this with

w(mi,mj), the purpose of this definition is explained below.

Remark 3.6.1.1. [Purpose of the definition]

1. Make D a linear function in n.

2. Divide the distance by the ratiomi+mj

m of multiplicities.

3. The bigger n is, the bigger D becomes.

4. The smaller w(mi,mj) is, the bigger D becomes.

3.6.2 Dispersion of VH and HH

In order to compute dispersion of a factor, two important things have to be prepared in advance.

Remark 3.6.2.1. [Prepare dispersion]

1. How to classify types (Classification)

2. How to compute d(vi, vj) (Distance)

Let’s define the conditions for VH and HH.

Definition 3.6.2.1. [Condition for VH]

1. The same t (the number of notes) and IH’s (Classification)

2. The difference of VH (Distance)

23

3.6. DISPERSION OF VH AND HH CHAPTER 3. MODELING

Definition 3.6.2.2. [Condition for HH]

1. The same t1, t2 (the numbers of notes) and IH’s (Classification)

2. The difference of HH (Distance)

Note that VHS and VHM have the same classification condition but different distance. Similarly,

HHS and HHM have the same classification condition but different distance. By these conditions, one

can define and compute the followings.

Definition 3.6.2.3. [Dispersion of VH and HH]

VHSD = dispersion of VHS

VHMD = dispersion of VHM

HHSD = dispersion of HHS

HHMD = dispersion of HHM

To summarize, 8 factors have been designed in this chapter. Classify them into two types.

Remark 3.6.2.2. [Types of factors]

Intensity factors

VHS, VHM, HHS, HHM

Dispersion factors

VHSD, VHMD, HHSD, HHMD

24

Chapter 4

Computer process

In the previous chapter, several concepts like PV, ID, IH, IRS were defined to design and compute

8 factors, which consist of 4 intensity factors (VHM, VHS, HHM, HHS) and 4 dispersion factors

(VHMD, VHSD, HHMD, HHSD) theoretically. However, everything can be computed by a computer

only when digitized data input is provided. This chapter will briefly explain how this study used

datasets for a computer.

4.1 Score to pitches

(a) The opening bar

(b) Raw data in Mi-crosoft Excel


Figure 4.1a is for the opening bar of Prelude in BWV 847. In the process of making digitized

datasets of figure 4.1b, IRS and pitches are the essential. Each row is for each IRS. For each IRS,

25

4.2. PITCHES TO IH CHAPTER 4. COMPUTER PROCESS

one has to write down all pitches which are sounding at that moment. In this study, this dataset was

made by using Microsoft Excel and pitches were saved with their names.

As mentioned in section 1.5, this study used two sets of musical works. The primary set isPreludes

in Well-Tempered Clavier, BWV 846-869, by Johann Sebastian Bach (1685-1750). The

secondary set is Preludes, Op. 37, by Giuseppe Concone (1801-1861). All scores of the sets are

from International Music Score Library Project (IMSLP)1. Two sets were converted to raw data files

of Microsoft Excel, that is, 48 musical works (24 by Bach, 24 by Concone) have their raw data files

separately. The conversion was done by hand for this study.

4.2 Pitches to IH

4.2.1 Pitches to PV

The definition of PV was introduced in section 3.1 so that the name of a pitch can be converted to

PV. This conversion was done in Microsoft Excel. Here is an example with the same opening bar used

in figure 4.1 and the value of PV.


4.2.2 PV to ID

The definition of ID was defined in section 2.3, and from now on, every conversion or computation is

done in Microsoft Visual Studio. Given two values of PV, let L be the difference of PV. The rule of

the conversion is simple by definition. If L ≤ 12, then it is done. If L > 12, compute L mod 12, and

the result should be 12 instead of 0 when L is divisible by 12. This rule coincides with the definition

of ID. The algorithm used in this study is shown as follows.

1http://imslp.org/

26

4.3. INTENSITY FACTORS CHAPTER 4. COMPUTER PROCESS

Figure 4.3: An algorithm of PV to ID

4.2.3 ID to IH

The conversion from ID to IH was defined in section 2.3, as definition 2.3.3.1 shows the conversion

rule. This trivial assigning function was defined in Microsoft Visual Studio. As a result, every interval

of two pitches can be converted to IH.

4.3 Intensity factors

By the result of section 4.1 and 4.2, it is now possible to compute VHM and VHS. Using the same

opening bar from figure 4.2, VH can be computed. In this example, every IRS has exactly two pitches,

allowing only one interval. As a result, figure 4.4b has one IH column and IH 1 is equal to VHS for

each IRS. In this case, VHM is equal to VHS for each IRS since every IRS has i2 =(22

)= 1.

(a) PV (b) VH


27

4.3. INTENSITY FACTORS CHAPTER 4. COMPUTER PROCESS

Now use another example which has more than two pitches for some IRS. One can understand

the following datatables with i1 = 1, i2 =(22

)= 1 and i3 =

(32

)= 3 for the number of IH. In addition,

remember that VHS is defined to be 4 when a specific IRS consists of one pitch.

(a) PV (b) VH


VHS and VHM for each IRS can be computed in this way. After computing VHS and VHM for a

whole musical work, it is possible to compute VHS per IRS and VHM per IRS.

Using the same example in figure 4.5, HH can be computed.

(a) PV (b) HH


HHS and HHM for each IRS can be computed in this way. After computing HHS and HHM for a

whole musical work, it is possible to compute HHS per IRS and HHM per IRS.

28

4.4. DISPERSION OF THE FOUR FACTORS CHAPTER 4. COMPUTER PROCESS

- VHM per IRS = (Sum of VHM) / (# of IRS)

- VHS per IRS = (Sum of VHS) / (# of IRS)

- HHM per IRS = (Sum of HHM) / (# of IRS - 1)

- HHS per IRS = (Sum of HHS) / (# of IRS - 1)

4.4 Dispersion of the four factors

Recall definition 3.6.2.1 and 3.6.2.2. Using these classification and distance for VH and HH, the goal

is to compute dispersion factors by definition 3.6.1.1 practically. Use the same example and observe

some parts of classification. Multiplicities are shown at the first column.

Figure 4.7: VH

Figure 4.8: HH

Now it is possible to compute VHSD of IRS, VHMD of IRS, HHSD of IRS, and HHMD of IRS

for a whole musical work. Consequently, given a musical work, this study can compute 8 factors with

4 intensity factors and 4 dispersion factors, using a computer. Remember that factors mean these 8

factors for the remaining part of this study.

29

Chapter 5

Analysis

One remark for this chapter is that intensity factors may omit ’per IRS’ (VHM means VHM per IRS,

and so on) and dispersion factors may omit ’of IRS’ (VHMD means VHMD of IRS, and so on.)

Recall three properties in section 1.5, which are the main targets in this study.

Major and minor

This study will show the difference of major and minor keys in several factors.

Key characteristic

From pure intonation and different frequency ratios of semitones, different major keys (minor

keys) had different characteristics. This study will check whether key characteristic does exist.

Key distance

This study will see whether the concept of key distance in the circle of fifths is realistic.

Note that the primary set of Bach and the secondary set of Concone are used as follows. Each

section will explain detail of analysis for each properties with the choice of sets.

Major and minor

Bach

Key characteristic

Bach, Concone

Key distance

Bach

Remember that IH was designed to have higher value when the intensity of a sound is higher,

according to Intensity Frequency Principle in section 2.3. In addition, IH is the element of 8 factors,

VHM, VHS, HHM HHS, VHMD, VHSD, HHMD, and HHSD.

30

5.1. MAJOR AND MINOR CHAPTER 5. ANALYSIS

5.1 Major and minor

First of all, let’s see the result of computation of 8 factors. Here are two tables of 12 Major and 12

minor, respectively, by Bach. Since the values of dispersion factors are too large, VHMD and VHSD

are scaled by 1100 , while HHMD and HHSD are scaled by 1

1000 .

Figure 5.1: Bach, Major

Figure 5.2: Bach, minor

Comparing the result of Major and minor, all factors show higher values for minor. To compare

Major with minor precisely, let’s see the table for the ratio of values of minor to values of Major.

Figure 5.3: Bach, Major and minor

Observe that all ratios are more than 100%, having 141% on the average. For intensity factors,

the average is 135%. For dispersion factors, the average is 147%. This result can be interpreted as the

following conclusion about Major and minor.

1. Harmony in minor is more intensive than harmony in major, with the ratio 135%

2. Harmony in minor is more dispersed than harmony in major, with the ratio 147%

31

5.2. KEY CHARACTERISTIC CHAPTER 5. ANALYSIS

5.2 Key characteristic

As mentioned at the beginning of this chapter, the secondary set of Concone is used only in this

section. Two sets of musical works will be compared to check whether they have a common pattern

of factors accoding to keys. If a common pattern does not exist in two different composers who lived

in different periods, it is hard to say that each key has its own original characteristic independent of

other things.

5.2.1 Graph and pattern

In the previous section, the computation of 8 factors was done for the primary set of Bach. The same

computation for the secondary set of Concone is done in this section, and the result for two sets will

be shown as a graph. Considering Major keys and minor keys respectively, each factor will have its

own graph to illustrate patterns of the factor for two sets together.

In order to see the pattern, a cycle of 12 major keys (minor keys) will be shown in a graph. To

do this, one of keys needs to appear twice as an initial point and a terminal point in a graph. Sicne

the choice of this key is not important, this study chooses C Major and A minor. Graphs and their

datatables will come in order as follows.

• Major, intensity factors

• Major, dispersion factors

• minor, intensity factors

• minor, dispersion factors

32


(a) VHM (b) VHS

(c) HHM (d) HHS

Figure 5.4: Major, Intensity factors (Blue : Bach, Orange : Concone)

Figure 5.5: Major, Intensity factors, Bach (Blue)

Figure 5.6: Major, Intensity factors, Concone (Orange)

33


(a) VHMD (b) VHSD

(c) HHMD (d) HHSD

Figure 5.7: Major, Dispersion factors (Blue : Bach, Orange : Concone)

Figure 5.8: Major, Dispersion factors, Bach (Blue)

Figure 5.9: Major, Dispersion factors, Concone (Orange)

34


(a) VHM (b) VHS

(c) HHM (d) HHS

Figure 5.10: minor, Intensity factors (Blue : Bach, Orange : Concone)

Figure 5.11: minor, Intensity factors, Bach (Blue)

Figure 5.12: minor, Intensity factors, Concone (Orange)

35


(a) VHMD (b) VHSD

(c) HHMD (d) HHSD

Figure 5.13: minor, Dispersion factors (Blue : Bach, Orange : Concone)

Figure 5.14: minor, Dispersion factors, Bach (Blue)

Figure 5.15: minor, Dispersion factors, Concone (Orange)

36


5.2.2 Analysis of pattern

It is ambiguous to analyze patterns of factors using only graphs. Note that the main task is to compare

patterns of Bach and Concone for each factor. To analyze and compare patterns, two methods are in-

troduced here. Suppose one factor has 12 values for Bach and Concone respectively. Let b1, b2, · · · , b12be the values of Bach and c1, c2, · · · , c12 be the values of Concone.

Definition 5.2.2.1. [0-100 Difference Mean (0-100 DM)]

1. Use notations bmin, bmax to denote the minimum and the maximum.

2. Note that all values are positive and bmin < bmax in this study.

3. Define b′t = 100 · bt−bminbmax−bmin

4. b1, b2, · · · , b12 is converted to b′1, b′2, · · · , b′12 as 0-100 values.

5. Do the same thing for c1, c2, · · · , c12 to obtain c′1, c′2, · · · , c′126. Compute the difference d′t = |b′t − c′t|7. DM = 1

12(d′1 + d′2 + · · ·+ d′12)

Definition 5.2.2.2. [Rank Difference Mean (Rank DM)]

1. Define r(bt) as the rank of bt among b1, b2, · · · , b122. Noth that all values are distinct in this study.

3. Compute ranks r(b1), r(b2), · · · , r(b12)4. Do the same thing for c1, c2, · · · , c12 to obtain r(c1), r(c2), · · · , r(c12)5. Compute the difference dt = |r(bt)− r(ct)|6. DM = 1

12(d1 + d2 + · · ·+ d12)

Let’s take a look at graphs and datatables of 0-100 scales and ranks in order as follows.

• Major, VH intensity factors

• Major, HH intensity factors

• Major, VH dispersion factors

• Major, HH dispersion factors

• minor, VH intensity factors

• minor, HH intensity factors

• minor, VH dispersion factors

• minor, HH dispersion factors

37


(a) VHM, 0-100 Scale (b) VHS, 0-100 Scale

(c) VHM, Rank (d) VHS, Rank

Figure 5.16: Major, VH Intensity factors (Blue : Bach, Orange : Concone)

Figure 5.17: Major, VH Intensity factors, Bach (Blue)

Figure 5.18: Major, VH Intensity factors, Concone (Orange)

38


(a) HHM, 0-100 Scale (b) HHS, 0-100 Scale

(c) HHM, Rank (d) HHS, Rank

Figure 5.19: Major, HH Intensity factors (Blue : Bach, Orange : Concone)

Figure 5.20: Major, HH Intensity factors, Bach (Blue)

Figure 5.21: Major, HH Intensity factors, Concone (Orange)

39


(a) VHMD, 0-100 Scale (b) VHSD, 0-100 Scale

(c) VHMD, Rank (d) VHSD, Rank

Figure 5.22: Major, VH Dispersion factors (Blue : Bach, Orange : Concone)

Figure 5.23: Major, VH Dispersion factors, Bach (Blue)

Figure 5.24: Major, VH Dispersion factors, Concone (Orange)

40


(a) HHMD, 0-100 Scale (b) HHSD, 0-100 Scale

(c) HHMD, Rank (d) HHSD, Rank

Figure 5.25: Major, HH Dispersion factors (Blue : Bach, Orange : Concone)

Figure 5.26: Major, HH Dispersion factors, Bach (Blue)

Figure 5.27: Major, HH Dispersion factors, Concone (Orange)

41


(a) VHM, 0-100 Scale (b) VHS, 0-100 Scale

(c) VHM, Rank (d) VHS, Rank

Figure 5.28: minor, VH Intensity factors (Blue : Bach, Orange : Concone)

Figure 5.29: minor, VH Intensity factors, Bach (Blue)

Figure 5.30: minor, VH Intensity factors, Concone (Orange)

42


(a) HHM, 0-100 Scale (b) HHS, 0-100 Scale

(c) HHM, Rank (d) HHS, Rank

Figure 5.31: minor, HH Intensity factors (Blue : Bach, Orange : Concone)

Figure 5.32: minor, HH Intensity factors, Bach (Blue)

Figure 5.33: minor, HH Intensity factors, Concone (Orange)

43


(a) VHMD, 0-100 Scale (b) VHSD, 0-100 Scale

(c) VHMD, Rank (d) VHSD, Rank

Figure 5.34: minor, VH Dispersion factors (Blue : Bach, Orange : Concone)

Figure 5.35: minor, VH Dispersion factors, Bach (Blue)

Figure 5.36: minor, VH Dispersion factors, Concone (Orange)

44


(a) HHMD, 0-100 Scale (b) HHSD, 0-100 Scale

(c) HHMD, Rank (d) HHSD, Rank

Figure 5.37: minor, HH Dispersion factors (Blue : Bach, Orange : Concone)

Figure 5.38: minor, HH Dispersion factors, Bach (Blue)

Figure 5.39: minor, HH Dispersion factors, Concone (Orange)

45


Here are datatables of 0-100 DM and Rank DM for major keys.

Figure 5.40: Major, 0-100 DM

Figure 5.41: Major, Rank DM

Figure 5.42: Major, 0-100 DM and Rank DM

Observe that 8 factors show the average value 30.96 of 0-100 DM, and the average value 3.25 of

Rank DM. Note that the minimum 0-100 DM is 24.62 of HHM per IRS, and the minimum Rank DM

is 2.17 of VHSD of IRS.

46


Here are datatables of 0-100 DM and Rank DM for minor keys.

Figure 5.43: minor, 0-100 DM

Figure 5.44: minor, Rank DM

Figure 5.45: minor, 0-100 DM and Rank DM

Observe that 8 factors show the average value 33.37 of 0-100 DM, and the average value 4.08 of

Rank DM. Note that the minimum 0-100 DM is 24.73 of VHMD of IRS, and the minimum Rank DM

is 3.17 of VHMD of IRS.

47


5.2.3 Conclusion

Let’s look at the result of 0-100 DM and Rank DM.

Figure 5.46: 0-100 DM

Figure 5.47: Rank DM

As explained before, 0-100 DM has average values 30.96 for major keys and 33.37 for minor

keys. For Rank DM, 3.25 is for major keys and 4.08 is for minor keys. In order to analyze this

result, a criterion is necessary to determine whether the value of DM is small enough to justify key

characteristic.

For the case of 0-100 DM, let’s compute the average distance when two numbers x and y vary

between 0 and 100. The average distance l =

∫ 100

x=0

∫ 100

y=0|x−y|dxdy

1002=

100

3 33.33 can be computed.

As a result, 0-100 DM values 30.96 (major) and 33.37 (minor) are not small enough to show significant

key characteristic.

For the case of Rank DM, compute a similar distance when two integers n and m vary from 1 to

12. The average distance l =12∑n=1

12∑m=1

|n − m| 1

122=

143

36 3.97 is not much bigger than Rank DM

values 3.25 (major) and 4.08 (minor), so that key characteristic is not significant.

One remark is that throughout all kinds of DM’s, major keys have obtained smaller values than

minor keys, meaning that key characteristic is relatively meaningful for major keys. Let’s summarize

this section with the following conclusion about Key characteristic.

1. 0-100 DM and Rank DM say that no significant key characteristic exists for either

major keys or minor keys.

2. Nevertheless, major keys have shown a little bit stronger key characteristic com-

pared to minor keys in 0-100 DM and Rank DM

48

5.3. KEY DISTANCE CHAPTER 5. ANALYSIS

5.3 Key distance

5.3.1 Pair in the circle of fifths

Recall the concept of the circle of fifths with the figure below.

Figure 5.48: The circle of fifths

Note that 12 major keys are covering the outer circle clockwise in order of C Major, G Major,

D Major, · · · , F Major, and again C Major. With this order, it is possible to number major keys

by N(key) so that N(C Major) = 0, N(G Major) = 1, N(D Major) = 2, · · · , and N(F Major) = 11.

Using this numbering, one can choose a pair of two major keys, and the number of all possible pairs

is(122

)= 12·11

2 = 66. In order to classify these pairs, define a t-pair as follows.

Definition 5.3.1.1. Two major keys, M1 and M2, form a t-pair if |N(M1)−N(M2)| ≡ t (mod 12)

For example, C Major and G Major form a 1-pair, and C Major and D Major form a 2-pair. Out

of 66 pairs, there are 12 1-pairs, 12 2-pairs, 12 3-pairs, 12 4-pairs, 12 5-pairs, and 6 6-pairs.

The same framework is used for 12 minor keys, with N(A minor) = 0, N(E minor) = 1, · · · , andN(D minor) = 11, with the silimar definition below.

Definition 5.3.1.2. Two minor keys, m1 and m2, form a t-pair if |N(m1)−N(m2)| ≡ t (mod 12)

49


5.3.2 t-DM of factors

There have been theoretical approaches to keys and their relation, such as graph and group in [6].

This study will set the circle of fifths as the minimum base for the relation of keys, with the use

of N(key) and t-pairs. Since major keys and minor keys share the same framework, it is enough to

explain everything for major keys. Given one factor, say VHM, and t-pairs, one can compute the

difference of VHM for each t-pair and the mean of all differences.

Definition 5.3.2.1. t-DM is the mean of all differences of a factor for all t-pairs.

Here are some examples.

Figure 5.49: Bach, Major, VHM

Figure 5.50: Bach, Major, t-DM of VHM

Figure 5.49 shows the values of VHM for 12 major keys by Bach, followed by figure 5.50 with the

differences of VHM and t-DM for each t-pair. This is how to compute t-DM for a given factor and

major/minor keys. Every factor can show its t-DM for major keys and minor keys.

50


Here are datatables for major/minor keys by Bach.

Figure 5.51: Bach, Major, t-DM

Figure 5.52: Bach, minor, t-DM

Recall that the goal of this section 5.3 is to check Key distance, the concept of ”close” keys

and ”distant” keys in the circle of fifths. The definition of t-pair for major/minor keys in the circle

of fifths is a suitable way. From a viewpoint of the circle of fifths, when two major keys, M1 and M2,

form a t-pair, it is reasonable to say that the bigger t is, the more distant they are from each other.

The same principle can be applied to minor keys.

Under this principle, if the concept of ”close” keys and ”distant” keys is realistic, t-DM should

increase as t increases. In other words, t-DM has to follow an increasing function of t. This study will

analyze datatables and graphs of t-DM to check it.

Let’s explain briefly how to check a correlation between t and t-DM. Suppose t-DM follows a

function f(t) of t. It means that one can get a horizontal line graph of t-DM/f(t). More details of

analysis will be explained with some examples.

51


Figure 5.53: Bach, Major, t-DM/1 of VHM

Figure 5.54: Bach, Major, t-DM/t of VHM

Figure 5.55: Bach, Major, t-DM/1 and t-DM/t of VHM (Blue : t-DM/1, Orange : t-DM/t)

Figure 5.53 is a datatable of t-DM/1 of VHM for 12 major keys by Bach, followed by figure 5.54

of t-DM/t. In this case, as seen in figure 5.55, the graph of t-DM is quite horizontal, meaning that

t-DM does not seem to increase as t increases. In other words, f(t) = 1 is more suitable than f(t) = t

for this case.

Like this example, the main task is to find a suitable f(t) which makes the graph of t-DM/f(t)

horizontal, that is, the graph of f(t) is similar to the graph of t-DM. When f(t) = 1 is suitable, it

means that the concept of ”close” keys and ”distant” keys is not realistic, as t-DM does not increase

as t increases. On the contrary, if f(t) is an increasing function of t such as f(t) = t or f(t) =√t,

the result will support the concept of ”close” keys and ”distant” keys, because the values of f(t) for

t = 1, 2, · · · , 6 will show an increasing shape of t-DM. First of all, datatables and graphs of t-DM/1

and t-DM/√t will be shown as examples in order as follows.

• Major, intensity factors

• Major, dispersion factors

• minor, intensity factors

• minor, dispersion factors

52


(a) VHM (b) VHS

(c) HHM (d) HHS

Figure 5.56: Major, Intensity factors (Blue : t-DM/1, Orange : t-DM/√t)

Figure 5.57: Major, Intensity factors, t-DM/1 (Blue)

Figure 5.58: Major, Intensity factors, t-DM/√t (Orange)

53


(a) VHMD (b) VHSD

(c) HHMD (d) HHSD

Figure 5.59: Major, Dispersion factors (Blue : t-DM/1, Orange : t-DM/√t)

Figure 5.60: Major, Dispersion factors, t-DM/1 (Blue)

Figure 5.61: Major, Dispersion factors, t-DM/√t (Orange)

54


(a) VHM (b) VHS

(c) HHM (d) HHS

Figure 5.62: minor, Intensity factors (Blue : t-DM/1, Orange : t-DM/√t)

Figure 5.63: minor, Intensity factors, t-DM/1 (Blue)

Figure 5.64: minor, Intensity factors, t-DM/√t (Orange)

55


(a) VHMD (b) VHSD

(c) HHMD (d) HHSD

Figure 5.65: minor, Dispersion factors (Blue : t-DM/1, Orange : t-DM/√t)

Figure 5.66: minor, Dispersion factors, t-DM/1 (Blue)

Figure 5.67: minor, Dispersion factors, t-DM/√t (Orange)

56


From the graphs and datatables, t-DM/1 looks more horizontal than t-DM/√t for most factors in

major/minor keys. In others words, t-DM does not seem to increase at a meaningful rate, and hence

the concept of ”close” and ”distant” keys cannot be supported by the result of t-DM.

1. For major and minor keys, key distance is not supported by t-DM of each factor.

2. ”Distant” keys in the circle of fifths do not show larger differences of 8 factors

(VHM, VHS, HHM, HHS, VHMD, VHSD, HHMD, HHSD).

5.3.3 t-DM of vectors

For a given musical work, this study computed 8 factors and analyze factors one by one. As shown,

the graphs and datatables did not support the concept of ”distant” keys. The next analysis will form

one vector for one musical works and use it to check Key distance. The first step is to convert 8

factors to 0-100 scale, as done in section 5.2, and the following datatables show the result.

Figure 5.68: Bach, Major, 0-100 Scale

Figure 5.69: Bach, minor, 0-100 Scale

57


With the datatables, one musical work can have one vector with 8 factors in 0-100 scale. Write the

vector V = (V HM,V HS,HHM,HHS, V HMD,V HSD,HHMD,HHSD) such as V (C Major) =

(63.71, 39.33, 0.00, 18.57, 12.92, 7.72, 8.71, 7.70). Add two more vectors Vi = (V HM,V HS,HHM,HHS)

and Vd = (V HMD,V HSD,HHMD,HHSD), so that Vi(C Major) = (63.71, 39.33, 0.00, 18.57) con-

sists of intensity factors and Vd(C Major) = (12.92, 7.72, 8.71, 7.70) consists of dispersion factors.

Now it’s time to compute the distance between two vectors. This study uses RMS as follows.

Definition 5.3.3.1. [Distance between two vectors]

- For V = (v1, · · · , v8) and V ′ = (v′1, · · · , v′8), d(V, V ′) =√

18

∑8k=1(vk − v′k)2

- For Vi = (w1, · · · , w4) and V ′i = (w′

1, · · · , w′4), d(Vi, V

′i ) =

√14

∑4k=1(wk − w′

k)2

- For Vd = (w1, · · · , w4) and V ′d = (w′

1, · · · , w′4), d(Vd, V

′d) =

√14

∑4k=1(wk − w′

k)2

Figure 5.70: Bach, Major, d(V, V ′) and t-DM

Figure 5.70 shows all 66 pairs for 12 major keys and their d(V, V ′), classified into t-pairs, in a way

similar with Figure 5.50 for t-DM of VHM for major keys. In this way, the following datatables will

be shown in order.

• Bach, Major, d(V, V ′) and t-DM

• Bach, Major, d(Vi, V′i ) and t-DM

• Bach, Major, d(Vd, V′d) and t-DM

• Bach, minor, d(V, V ′) and t-DM

• Bach, minor, d(Vi, V′i ) and t-DM

• Bach, minor, d(Vd, V′d) and t-DM

58


Figure 5.71: Bach, Major, d(V, V ′) and t-DM

Figure 5.72: Bach, Major, d(Vi, V′i ) and t-DM

Figure 5.73: Bach, Major, d(Vd, V′d) and t-DM

59


Figure 5.74: Bach, minor, d(V, V ′) and t-DM

Figure 5.75: Bach, minor, d(Vi, V′i ) and t-DM

Figure 5.76: Bach, minor, d(Vd, V′d) and t-DM

60


Here is a datatable for t-DM of d(V, V ′), d(Vi, V′i ), and d(Vd, V

′d) for major/minor keys.

Figure 5.77: Bach, t-DM of vectors

From the datatable, t-DM does not seem to increase at a meaningful rate for every type of vectors

in major/minor keys, and thus the concept of ”close” and ”distant” keys cannot be supported by the

result of t-DM.

1. For major and minor keys, key distance is not supported by t-DM for vectors.

2. ”Distant” keys in the circle of fifths do not show larger differences of vectors (V, Vi, Vd).

5.3.4 Conclusion

Based on t-DM of factors and vectors, ”distant” keys do not show significant difference compared to

”close” keys, because t-DM does not increase at a meaningful rate as t increases. Let’s summarize

the result with the following conclusion about Key distance.

1. t-DM of factors and vectors say that the concept of ”close” keys and ”distant”

keys in the circle of fifths is not supported by the result of computation.

2. Regardless of major keys or minor keys, key distance did not appear to be realistic

for any factor or vector.

61

Bibliography

[1] 전상직 (2014) The Principle of Music (음악의 원리), 음악춘추

[2] William A. Sethares (2005) Tuning, Timbre, Spectrum, Scale (Second Edition), Springer

[3] 백병동 (1998) 화성학 (증보판), 수문당

[4] William A. Sethares & Ryan Budney (2014) Topology of musical data, Journal of Mathematics

and Music, 8:1, 73-92, http://dx.doi.org/10.1080/17459737.2013.850597

[5] William M. Hartmann (2013) Principles of Musical Acoustics, Springer

[6] Adrian Walton (2010) A graph theoretic approach to tonal modulation, Journal of Mathematics

and Music, 4:1, 45-56, http://dx.doi.org/10.1080/17459730903370940

국문 초록

이 연구는 음악의 조와 화성을 분석하는 것을 목표로 하였다. 바흐의 작품이 주된 소재로 사용되었다.

바흐의 작품을 사용하면서 분석의 주된 대상은 조의 3가지 특성이었다. 장조와 단조 는 장조와 단조의

차이를 발견하는 것을 말한다. 조의 특색 은 각 장조와 단조의 특색에 관한 것을 말한다. 조의 거리 는 조

사이의 이론적인 거리를 말한다.

조율법은 음의 진동수를 결정하고 화성에 영향을 준다. 진동수 비는 이 연구에서 화성을 계산하는

데에 기초가 되는 내용이다. 음악 작품을 분석하여 조의 3가지 특성을 확인할 수 있도록 의도적으로 진동

수 비를 설정하였다. 그런 진동수 비를 사용하여 수직적 화성과 수평적 화성을 각각의 음악 작품에 대해

계산하였고, 한 작품 안에서 화성의 분산 역시 정의하고 계산하였다.

계산 결과를 통해 조성의 3가지 특성에 대해 결론을 내릴 수 있었다. 장조와 단조는 장조에 비해

단조가 35% 더 강한 화성을 사용하고 47% 더 화성이 분산됨을 통해 확인되었다. 조의 특색은 장조와

단조모두에서의미있는특색을가지기에는약하다는결론을내렸다.조의 거리는계산결과로입증되지

않음으로써 현실적이라고 볼 수 없었다.

주요어 : 바흐, 조율법, 진동수 비, 조, 화성

학 번 : 2015-20252

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