FRANCIS SINCO Math 290

THE MATHEMATICS OF MUSICAL MODES ANDHARMONICS

by

Francis Brylle Sinco

A graduate research paper

submitted to the

Institute of Mathematics

College of Science

The University of the Philippines

Diliman, Quezon City

in partial fulfillment of the

requirements for the degree of

Master of Arts in Mathematics

December 2014

The University of the Philippines

College of Science

Institute of Mathematics

This is to certify that this graduate research paper entitled

“The Mathematics of Musical Modes and Harmonics”,

was submitted by Francis Brylle Sinco to fulfill part of the requirements for the degree of Master

of Arts in Mathematics.

Fidel R. Nemenzo, D.Sc.

Adviser

Abstract

The Mathematics of Musical Modes and Harmonics

Francis Brylle Sinco Adviser:University of the Philippines, Diliman Fidel R. Nemenzo, D.Sc.December 2014

The disciplines of Math and Music are both built around patterns, cycles, and the inherent

relationship between quantities, as we often see in mathematical formulas and musical chords and

scales. Foundations of the theories of Math and Music often trace their roots in the same people

such as Pythagoras and Euclid, among others. In this paper we shall establish and explore the

innate links between the two major disciplines. In particular, we shall focus on the discovery of

(string) harmonics by Pythagoras and the various modes of both Eastern and Western music (with

our discussion concentrating on the latter). And finally, we will translate the language of musical

modes, scales, chords, and harmonics into the language of Math using sets, sequences, and networks.

iii

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Mystic and Scientific . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Law of Octaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 A Great Fascination with Ratio

and Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 The Other Musical Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 The Equal-Tempered Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Chapter 2. Rudiments of Music Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Major Triad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Minor Triad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.3 Other 3-Chords and 4-Chords Derived from the Major Triad . . . . . . . . . 14

2.2.4 Harmonic Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Major and Minor Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2 Other Scales in Various Cultures . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 The Seven Modern Greek Modes of Music (18th century onwards) . . . . . . . . . . 19

Chapter 3. Chords, Scales, and Modes:

A Mathematical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Cocos and Fowers (2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 The Marriage of Music and Math:

Reconciling Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Patterns and Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Chapter 4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

iv

Chapter 1

Introduction

Math and Music are deemed to be languages on their own. In fact, ancient Greeks even believed

that these are the language with which the Universe is built. Unlike other disciplines, they both were

not invented but rather, merely discovered, explored, unlocked, and then progressively systematized

into organized sciences. In the old times, they were just both abstractions about – but not limited

to – numbers/figures/patterns and sound, respectively, and not the organized bodies of knowledge

we know today. That is, the respective theories of Math and Music were borne out of abstractions

mainly about patterns - with Math concerning patterns of numbers (and again, not just numbers)

and Music being about sounds (in particular, about which sounds blend well together and which do

not, at least in the standards of their era).

In this text, we shall refer to the organized disciplines or bodies of knowledge as “Math” and

“Music”, written as such – with capitalized first letters – and we shall write “math” and “music”

for all other uses of the terms.

1.1 Mystic and Scientific

While officially, theoretical musical notation (solfege) was invented by an Italian monk by the name

Guido di Arezzo (or d’Arezzo by contraction) more than a millennium after the Greeks first formed

crude thoughts about Music Theory, the origins of Music Theory itself could not be attributed to a

single person in history. Rather, it was – and is – a conglomeration of concepts arising from different

periods. That is, at least the (Western) Music Theory we know now. However, one prominent figure

from around the 6th century BCE is credited for a number of important elements in Modern Music

Theory. Pythagoras, the great Greek metaphysics philosopher and the man we all recognize for his

notorious theorem on right triangles, is dubbed by a number of scholars as the Father of Modern

Music Theory. Now it should be no surprise that Pythagoras was more of a household name in

Math than in Music; however it should be noted that he was also the great thinker behind the very

foundations of Music Theory. He made profound observations about the sounds strings of varying

lengths made. And that precisely formed the foundations of the study of musical intervals and

harmonics.

Besides scientific observations on geometry and music, he also formed some great questions about

1

Chapter 1. Introduction 2

the Universe. Little do people know that in his time, he stood to unite Math, Science, and Mysticism.

In southern Italy (in a place called Crotona), he founded his own school - the Pythagorean Academy

- which began as a commune of about 300 people dedicated to academic and mystical pursuits.

While his contemporaries scorned at the idea of mixing the scientific and the mystic, he did not view

his endeavors as contradictory. One of their – the Pythagoreans, as they used to call themselves –

principal teachings was that,

“God is universal harmony, perceived through number.”

His contributions to geometry were a result of his extensive sojourn in Egypt and Babylonia.

However numerous and far-reaching his and his school’s legacy have been, they did not leave any

written accounts since, allegedly, they intended to pass on their teachings purely by oral recitation

and only to a select group of people. That is why everything that we now perceive to be truths

about Pythagoras and his school were only patches of history told by various scholars from different

periods.

Among these purely oral teachings was the great idea of the so-called Music of the Spheres.

Pythagoras taught that each of the 7 Classical Planets produced a “note” upon revolving around

its orbit about a fixed center – which was our Earth (and as we see here, Pythagoreans adhered

to Ptolemy’s Geocentric Model of the Universe). Only that this “note” is so fine, our human ears

could not possibly hear it; that we could only hear this sound once we achieve a certain heavenly

state, possibly beyond this earthly existence. Others even believed that this sound could well be

some kind of mathematical or mystical concept.

We shall note here, however, that the 7 Classical Planets of their time were not similar to the

planets we officially recognize now. Instead, in antiquity, people used to refer to the following

celestial bodies as the 7 Classical Planets in the belief that they revolve around us here on Earth:

Sun, Moon (Luna), Mars, Mercury, Jupiter, Venus, and Saturn. The order they are presented here

pertains, in fact, to the way the 7 Gregorian days are lined up. These 7 Classical Planets are actually

namesakes of the 7 days of the week (originally in Latin, but we shall present them here in their

Spanish counterparts for ease of reference; the same applies to the other Romance languages and

their variations, including Italian, Esperanto and Filipino): Sun - domingo, Moon (Luna) - lunes,

Mars - martes, Mercury - miercoles, Jupiter - jueves, Venus - viernes, Saturn - sabado.

1.2 The Law of Octaves

The Music of the Spheres conjecture was probably the precursor of the concept of a Diatonic Scale

(a scale with 7 distinct notes, which repeat thereafter, only in a higher octave) in Music, such as the


Figure 1.1: Musica Mundana

Major Scale and the Minor Scales (Natural, Melodic, and Harmonic Minors) which are ubiquitous

in almost all kinds of music, in general. This concept of “music” produced by the 7 Classical Planets

was called Musica Mundana (literally, Music of the Spheres).

Physically real or not, this Music of the Spheres idea had great attributions to The Great Reli-

gions. It was believed that Moses heard it upon receiving the tablet bearing the Ten Commandments

on Mt. Sinai, Egypt. Saint Augustine also believed that this is the sound people hear at the thin

moment separating life from worldly death. Moreover, the rhythm of this celestial music was also

believed to govern the cycles of seasons and other processes of life.

In Science, more importantly, we could attribute this concept to a very remarkable observation.

Scientists even have a name for this: Law of Octaves. This idea could very well be related to the

Music of the Spheres, in the conjecture that some things in nature follow the pattern that after

the seventh element or term, the cycle repeats. One concrete example of a scientific phenomenon

following the Law of Octaves manifests in the way the chemical elements behave. It was observed by

J.A.R. Newlands in 1865 that elements, when arranged in increasing atomic weight, tend to repeat

the same physical and chemical properties every interval of seven elements.


1.3 A Great Fascination with Ratio

and Proportion

Pythagoras upheld the belief that the Universe was governed by mathematical rules. In particular,

he believed that everything could be explained using the relationship between numbers, i.e., ratios

and proportions. This is evident in his explanations of the nature of sounds – those pairs or groups

of sounds that blend well (“consonant notes”) and that do not (“dissonant notes”).

However, it should be noted that this subjectivity on “pleasantness” of note combinations is

purely a Western construct, in particular, Greek. That is, what sounded dissonant to ancient

Greeks could possibly have sounded consonant to ancient Indians or Chinese, and vice versa. In our

discussions on harmonics, for simplicity and by default – and with reference to the current Western

Music Theory standards – we shall stick to the Western perspectives on consonance and dissonance

unless otherwise stated. However, one should also note that this shall by no means discriminate

between the music of the East and the West.

Legend has it (and legend indeed, since the Pythagoreans did not leave a single tangible trace)

that one time Pythagoras happened to pass by a blacksmith’s and there he saw and heard two

anvils of different sizes – one twice the other – being hammered and producing consonant notes

(“consonant,” at least in his perspective). Since the bigger anvil is a whole-number (i.e., 2) multiple

of the smaller one in terms of size, naturally he pondered if there exist some other anvils in sizes

between theirs, which could also sound in consonance with any of them. Translated mathematically,

the inquiry pertains to the existence of an anvil or anvils a fraction bigger than the smaller one

which would sound in consonance with them when struck or hit.

He then shifted his inquiry into strings, which, apparently, are by far cheaper and more practical

and convenient to manipulate than anvils in terms of size and length alterations. He discovered

that if you get two strings, tie them taut, and then press or muffle one of them2

3of the way, they

also produce consonant notes. And having great fascination with patterns and special and beautiful

mathematical relationships, he also tried pressing one of the strings3

4of the way, and to his huge

amazement they also sounded consonantly.

We shall remark that in his time, they did not have the concept of written notes, or even a system

for naming them (as it would only be due to Guido d’Arezzo more than a millennium forward). No

C − D − E − F − G − A − B − C+ or do-re-mi-fa-sol-la-ti/si-do, or flats and sharps and clefs.

They only relied on the relationship between any two sounds or what we now refer to as “musical

intervals”.

Instead, Pythagoras called such 1 : 2 relationship between big and small (whether anvil or string)

as a Diapason; the 2 : 3 as Diapente; and the 3 : 4 as Diatessaron. Notice the elegant 1− 2− 3− 4


Figure 1.2: Pythagorean Strings

telescopic pattern. In today’s terms, we call them Octaves, Perfect Fifths, and Perfect Fourths,

respectively.

Hence, the notion of the three very common consonant musical intervals of the Pythagoreans

was born. Eventually in the tradition of Western Music Theory, some other intervals would become

accepted as consonant – the Thirds and the Sixths – as we shall see further in this discussion.

1.4 The Other Musical Intervals

So far we have discussed the origins of the concepts of octaves (1 : 2), perfect fifths (2 : 3), and

perfect fourths (3 : 4). That is, in terms of strings, if we tie taut strings with lengths 1 unit and 2

units and pluck them, the resulting notes are an octave apart; strings 2 units and 3 units long will

produce perfect fifth notes; and strings 3 units and 4 units long will produce perfect fourth notes.

We could see the telescopic pattern of n : (n+ 1) there, and the most natural inquiry would be

about the existence of any other value of n which would produce consonant notes, least in Western

perspective. Again, in Pythagoras’s time the concept of the twelve acoustically-distinct chromatic

notes (as in the piano keys) we perceive today were unbeknownst to the world. So there could have

been no way back then to exactly illustrate the musical interval notations we use today.

And in fact in the evolution of taste and perception in Music, people started to accept some

other intervals following the n : (n + 1) pattern as consonant. But following these beautiful but

limiting Pythagorean ideal ratios and proportions actually proved problematic upon the institution

of the 12-note Chromatic Scale {C,C#/Db,D,D#/Eb,E, F, F#/Gb,G,G#/Ab,A,A#/Bb,B} a

few centuries later. That is, following the Pythagorean ideal ratios and proportions, there would be

notes that are theoretically consonant but actually dissonant when played on an instrument.

This problem springs from the fact that while it is theoretically possible to fill in the space

between an arbitrary starting note (called “fundamental note” or “tonic”) and its perfect fourth

(3 : 4) with notes that follow the n : (n+ 1) pattern (4 : 5, 5 : 6, 6 : 7, 7 : 8, 8 : 9, et al.) with respect


to that arbitrary starting note, they may not be actually consonant as per the whole standards of

Western music.

That being said, when Pythagorean ratios are applied to the 12-note Chromatic Scale (in any

starting note, or in musical jargon, “key”) the intervals would not be equally divided. It is quite

apparent because we cannot really partition 1 : 12 into twelve n : (n+ 1) ratios, with n ∈ N. Hence,

modern music theorists have devised a method to accommodate all twelve notes of a full Chromatic

Scale (the scale that exhausts all notes) while still preserving approximately the 1 : 2 (octaves), 2 : 3

(perfect fifths), and 3 : 4 (perfect fourths) original and crude Pythagorean ratios.

This tweaking was done purely mathematically by treating the twelve distinct notes as points on

the interval [x, 2x], where x > 0 – without loss of generality – rather than on a set of specific fractions

n : (n + 1) from x to 2x. That is, each of the 12 distinct musical notes could then be assigned an

irrational value which denote its relative distance (wavelength) from the arbitrary starting note, or

“tonic”. The tonic would then be assigned a virtual “string length” x. And since there are going

to be twelve notes on a full octave, then each pair of successive notes would have a String Length

Ratio of

(2x

x

) 112

=

(2

1

) 112

= 2112 (1.1)

For instance, in the C Major Scale {C,D,E, F,G,A,B}, the interval from C to D, which is a

whole tone or two semitones (“half tones”) in length, would have a ratio equivalent to 2212 . And from

E to F, just a semitone, or 2112 . And indeed, the Pythagorean ideal interval ratios are approximately

preserved.

• C to F (Perfect Fourth, 3 : 4)

5 semitones or 2512 ≈ 1.3348 ≈ 1.3333 =

4

3• C to G (Perfect Fifth, 2 : 3)

7 semitones or 2712 ≈ 1.4985 ≈ 1.50 =

3

2• C to C+(Octave, 1 : 2)

12 semitones or 21212 = 2 =

2

1

This tweaked rule applies to just any other choice of tonic in the Chromatic Scale since any two

consecutive notes there are 2112 “string lengths” apart.

This modern tuning for notes is called the Equal-Tempered Scale. Intuitively its name suggests all

semitone intervals are identical – both mathematically (ratio) and musically (sound) – regardless of


the scale type. A standard semitone ratio, or temperament, is set at 2112 , and in general a k-semitone

interval would bear a ratio of 2k12 between the notes.

Now in terms of engineering the musical instruments, the manufacturers and tuners typically

would need a more convenient notation, devoid of any fractional elements, just as we prefer with

currency and denominations. That is why modern music theorists further devised a more convenient

quantification scheme for intervals, which is based on the equal-tempered scale.

1.5 The Equal-Tempered Scale

In the equal-tempered scale, they called each unit a “cent” (and again, like monetary units) and

proposed that each semitone be composed, conveniently, of 100 cents. So that one full octave bears

a total of (100 cents/semitone)*(12 semitones) = 1200 cents.

With respect to the 2112 rule for a semitone, each cent would then be equal to a hundredth root of

a semitone (since 1 semitone = 100 cents). Hence we have the following cent-semitone relationship:

1cent = (1semitone)1

100

=(

2112

) 1100

= 21

1200 (1.2)

So that

1octave = (1cent)1200

=(

21

1200

)1200= 2

12001200

= 2 (1.3)

just as expected.

Moreover, it goes without saying that a k-semitone interval would be equivalent to 100k cents.

Now let us take a look at an example:


Table 1.1: Equal-Tempered Scheme applied to the C Major Scale

NOTE SEMITONES (k) 2k12 CENTS

C 0 2012 = 1 0

D 2 2212 200

E 4 2412 400

F 5 2512 500

G 7 2712 700

A 9 2912 900

B 11 21112 1100

C+ 12 21212 1200

This equal-tempered scheme is currently (and has long been) the standard for tuning in Western

music. Equivalently in terms of Physics, there are certain frequencies (in hertz, Hz) conveniently

assigned to each note.

Furthermore, the following are the common names for the intervals with respect to the tonic.

Some intervals bear more than one name, but here we shall just list the commonly used name for

each. We also label which are consonant intervals, at least in Western perspective.

Table 1.2: Common Names for Intervals (illustrated in C Chromatic Scale)

NOTE INTERVAL SEMITONES (k) CENTS CONSONANT ?C (Tonic) Unison 0 0 N/ADb (C#) Minor Second 1 100 NoD Major Second 2 200 NoEb (D#) Minor Third 3 300 Yes (modern)E Major Third 4 400 Yes (modern)F Perfect Fourth 5 500 Yes (Pythagorean)F#/Gb Middle Tone 6 600 NoG Perfect Fifth 7 700 Yes (Pythagorean)Ab (G#) Minor Sixth 8 800 Yes (modern)A Major Sixth 9 900 Yes (modern)Bb (A#) Minor Seventh 10 1000 NoB Major Seventh 11 1100 NoC+ Octave 12 1200 Yes (Pythagorean)


The intervals bearing 3, 4, 8, and 9 semitones were a modern addition to the original Pythagorean

consonant intervals (5, 7, and 12 semitones).

1.6 Summary

In conclusion, we saw how the primeval theories of music gradually evolved from simple mathemat-

ical inquiries and observations involving aesthetic in numbers (1 : 2, 2 : 3, 3 : 4) by Pythagoras.

Although the Pythagorean theoretical scheme for intervals are not fully implemented in today’s

music due to the practical dissonance of some pairs of notes, we owe the concept of perfect fourths,

perfect fifths, and octaves to him and his school. These are fundamental concepts in Music that

form the basis of our Modern Music Theory.

*****************

In this paper we shall concentrate on the musical foundations by Pythagoras. Unless otherwise

stated, we will be dealing with Western perspectives on Music. At some points in the course of the

discussion, however, we could introduce some Eastern influences for variety and enrichment.

Lastly, our main goal in this paper is to investigate the inherent mathematical structures present

in musical scales, in particular in the consonant notes. In the latter part of the study we shall

illustrate geometrically how “pleasant sounds” look like. Since in the majority this is a mathematical

study, we should also aim to define existing musical notions in modern mathematical terms, and also

form compact generalizations. Hence, although we will be showing examples of scales and chords

from time to time, much of the discussion shall focus in the arbitrary language of mathematical

variables and sets.

We shall see that a family of equivalent scales (e.g., C Major, D Major, et al.) share the same

graph or network, and that other scales of the same number of distinct elements could be formed

out of one graph using basic transformations.

And by these generalized sets and networks for various families of scales we aim to understand

the inherent mathematical patterns in Music, whose beauty inspired Pythagoras in forming the

foundations for Modern Music Theory.

Chapter 2

Rudiments of Music Theory

In this chapter we shall explain the rudiments of Music Theory which are relevant to the mathe-

matical topic under discussion. In particular, we shall focus on the concepts of musical intervals,

chords, scales, and modes.

We just saw how the basic ideas of the perfect fourths, perfect fifths, and octaves originated from

the great metaphysics philosopher, mystic, and mathematician Pythagoras.

We also diverted our discussion towards the standardization of a semitone using “cents” which

indicated a little deviation from the Pythagorean semitones, but nonetheless approximately preserves

the original interpretations of the perfect fourths, perfect fifths, and octaves. These “cents” are so

convenient and useful that they are employed in orchestral tuning following today’s Western music

standards.

Hence hitherto we shall adhere to the cent-based definition of semitones so that each semitone

is of standard “string length” of 2112 units, and that an octave bears a full 2 “string length” units.

And from here, we can proceed with defining the related and necessary concepts in Music Theory

in preparation for succeeding discussions.

In this chapter, we shall present the musical definitions of intervals, chords, scales, and modes.

That is, before presenting them into a unified mathematical concept, we shall first investigate their

existing definitions in Music. We shall also make brief historical notes about scales and modes since

each culture apparently tends to have its own system of music.

For the purpose of discussion, in this chapter we opt to focus on the purely musical perspectives

on the subject. Now it would be necessarily assumed that the reader has at least the basic knowledge

of musical notations (solfege, accidentals, key signatures, clefs) and common scales (at least the C

Major Scale).

2.1 Intervals

A musical interval basically gives the number of semitones between two notes (in ascending order).

For instance, the interval from C to D consists of two semitones. That is, one semitone from C

to C#/Db, and then another from C#/Db to D. In general, we call a 2-semitone interval a major

second. Note that the interval from D to C+ is another matter.

10

Chapter 2. Rudiments of Music Theory 11

However, some certain number of semitones bear multiple names. It all depends on which letter

we use to represent the inclusive notes (e.g., just as in the case of C# or Db, which are essentially

the same). This has something to do with the concept of enharmonic notes.

Enharmonic notes are two notes that bear different names, but are essentially the same notes in

terms of sound (frequency). In Western music, the Chromatic Scale serves as the universal set for

scales, i.e., it contains all the notes possible (a total of 12 notes). Inside this scale, the enharmonic

notes are: C#/Db, D#/Eb, F#/Gb, G#/Ab, and A#/Bb (sometimes and in very rare, almost

nonexistent, circumstances, E#/F,Fb/E B#/C, and Cb/B appear).

The main purpose for the existence of enharmonic notes is to retain the consistency among the

letters and “formula” characteristic to a scale. For instance, in the A Major Scale

{A,B,C#, D,E, F#, G#}, if we use any or all of Db, Gb, and Ab, in place of C#, F#, and G#,

respectively, then there would be duplication of letters (it would look like {A,B,Db,D,E,Gb,Ab}).Another example would be in triads (3-note chords). In the Cm (C minor) Triad {C,Eb,G}, if we

replace Eb by D#, the usual ubiquitous C-E-G combination would be tampered with.

In some cases, however, the repetition of letters (e.g., G and Gb) is inevitable, as in the Blues

Scale. For example, in the C Blues Scale {C,Eb, F,Gb,G,Bb}, whether we use F# or Gb, there

would still be a repetition of letters, since the accidental (Gb), and F and G are all present. Hence,

the idea of enharmonic notes serves only the purpose of preserving the “consistency” in nomenclature,

whenever possible, and has nothing to do with any mathematical and musical aspects of sound.

However, enharmonic notes are the reason behind the multiple names for otherwise fundamentally

one and the same intervals. For instance, the 1-semitone intervals C-C# and C-Db are one and the

same. However, the notion of enharmonic notes dictates that they be named an augmented unison

and a minor second, respectively.

The following table illustrates the intervals from the tonic C (as illustrated in the C Chromatic

Scale) including all enharmonic notes.


Table 2.1: An Extensive List of Intervals (illustrated in C Chromatic Scale)

NOTE SEMITONES (k) INTERVAL NAMEC (Tonic) 0 UnisonC# 1 Augmented UnisonDb Minor SecondD 2 Major SecondD# 3 Augmented SecondEb Minor ThirdE 4 Major ThirdF 5 Perfect FourthF# 6 Augmented FourthGb Diminished FifthG 7 Perfect FifthG# 8 Augmented FifthAb Minor SixthA 9 Major SixthA# 10 Augmented SixthBb Minor SeventhB 11 Major SeventhC+ 12 Octave

Note that the perfect fourth, perfect fifth, and octave still bear their special status, as they had

with Pythagoras. From now on we shall denote an octave note by the tonic note with a plus (+) to

indicate that it is a full octave above the tonic. And also, the terms “augmented” and “diminished”

simply mean raised (#) or lowered (b) by a semitone, respectively.

Although at first the table might seem a little too convoluted due to the presence of redundant

intervals, in our discussion we shall just focus on the number of inclusive semitones and not the

interval names, to retain arbitrariness.

Lastly, in the previous chapter we took a glimpse of the concepts of consonance and dissonance.

Inside a full octave, consonant notes have 3, 4, 5, 7, 8, or 9 inclusive semitones; otherwise, they are

called dissonant notes.

2.2 Chords

Loosely speaking, any combination of at least two notes is called a chord. But since there will be

trivially many of these kinds, we shall just limit our definition of a chord to combinations that have

some special musical significance.

Chords are characterized by the way they build harmony in a musical composition. They are the

basic foundations of the structure of music; some chords are even added to inject a certain flavor or


character to an otherwise plain and predictable piece of music. Sometimes just tweaking one note

in a chord could change the mood of a passage altogether.

In terms of Music Theory, we will denote a p-chord by a sequence with p elements. For instance,

the C Major Triad (a 3-chord) will be denoted as {C,E,G}. Following this scheme, the Interval

Sequence corresponding to a chord is the sequence of p elements containing W’s (whole tone), H’s

(half tone), or a linear combination of them, representing the “spaces” or intervals between these

p elements, plus the interval from the pth chord note to the octave note. Thus, the corresponding

Interval Sequence of the C Major Triad – or any Major Triad, for that matter – is given by {2W,

H+W, 2W+H}. That is, there are 2 whole tones (2W) from the tonic (C) to the major third (E),

a tone-and-a-half (H+W) from there to the perfect fifth (G), and then finally two-and-a-half-tones

(2W+H) from the perfect fifth to the octave note (C+). In fact, the Interval Sequence representation

of a certain type of chord, which is tonic-invariant, is more important than the chord (with its notes

indicated), itself.

So from now on, in the Interval Sequence representations of chords and scales, we shall break

down intervals into basic units of whole tones (W = 2 half tones) and half tones (H = 1 half tone).

Any musical chord or scale (or mode, as we shall see soon) can be expressed in terms of them, or a

combination of them.

Since basically, a chord or scale is just a partitioning of intervals in a “string” with 12 semitones

(or, 12 H’s), then the sum of all the elements in the Interval Sequence of ANY CHORD

OR SCALE (or MODE) is always 12, given W = 2 and H =1.

Moreover, we shall limit our discussion to 3-chords (“Triads”) and 4-chords.

2.2.1 Major Triad

{2W, H+W, 2W+H}Perhaps the most common type of chords due to their “standard” flavor and versatility of sound,

these chords are the staples of Western music.

These chords are the main building blocks of Western harmony. They are so fundamental that

all chords can be derived from them using accidentals and/or addition of notes.

A 3-chord type, these chords are characterized by the inclusive Major Third (2W), Minor Third

(H+W), and Perfect Fourth(2W+H) intervals, in that order. In general, a Major Third interval

sounds bright, while a Minor Third interval sounds heavy and gloomy. So it is this positioning of

Major and Minor Thirds that make it a Major Triad.

To build a Major Triad, start with any root or tonic. Then add the Major Third above the tonic,

and from there add the Minor Third of the second note. For instance, the C Major Triad is built by

starting at the tonic C, then moving on to E (Major Third above C), then moving on to G (Minor


Third above E), and finally capping it off by ending at the octave C+ (Perfect Fourth above G). So

the C Major Triad is written {C,E,G}. As a result, the third note is at the same time a Perfect

Fifth above the tonic, and a Perfect Fourth below it (but from the next octave). Another example

would be the Ab Major Triad {Ab,C,Eb}.

2.2.2 Minor Triad

{W+H, 2W, 2W+H}This relative of the Major Triad is constructed by switching the position of Minor and Major

Thirds. The resulting chord is a heavier, darker version of the Major Triad. The C Minor Triad (or

Cm) is given by {C,Eb,G}. since we just switched the inclusive intervals, this alteration is simply

equivalent to ”flattening” the second note without changing the tonic and the third note.

2.2.3 Other 3-Chords and 4-Chords Derived from the Major Triad

As we have previously noted, a multitude of other chords arise from modifications of the Major Triad.

These special chords are employed by musicians in order to precisely capture the intended mood

of the composition, which could not be exactly expressed using just the Major and Minor Triads.

A lot of these are extensively used in Jazz, Blues, Rock, Spanish, Eastern (Asian), Indian, Middle

Eastern, Eastern European, Greek, and Gypsy music, among others. Here we shall present them to-

gether with their respective inclusive intervals. For illustration purposes we will be using the tonic C.

Table 2.2: Some 3-Chords and 4-Chords (illustrated in C Chromatic Scale)

CHORD INTERVAL SEQUENCE CHORD IN CAugmented {2W, 2W, 2W} {C,E,G#}Diminished {W+H, W+H, 3W} {C,Eb,Gb}Major 7th {2W, H+W, 2W, H} {C,E,G,B}Dominant 7th {2W, H+W, W+H, W} {C,E,G,Bb}Minor 7th {W+H, 2W, W+H, W} {C,Eb,G,Bb}minor/MAJOR 7th (miMA7th) {W+H, 2W, 2W, H} {C,Eb,G,B}minor 7th, flat 5th {W+H, W+H, 2W, W} {C,Eb,Gb,Bb}Diminished 7th {W+H, 2W, W, W+H} {C,Eb,G,Bbb(A)}Major 6th {2W, W+H, W, W+H} {C,E,G,A}Minor 6th {W+H, 2W, W, W+H} {C,Eb,G,A}


2.2.4 Harmonic Chords

Since we are dealing with the mathematical concepts of Music, we shall now introduce the concept

of harmonic chords. These are chords whose notes are in mutual consonance with each other. That

is, if we take any note from the chord, its interval with any other note in the chord (both ascending

and descending) has 3, 4, 5, 7, 8, or 9 semitones.

Cocos and Fowers (2011) proved that a harmonic chord can have only two (the trivial case) or

three distinct notes (triad), considering there is a maximum of 12 semitones in any interval. An

addition of a fourth (distinct) note will put the mutual harmony into disarray, at least mathematically

(since in practice, there are pleasant 4-chords, 5-chords, and so on). We shall discuss their proof

briefly in the next chapter.

Let us take, for instance, the C Minor 7th chord, a 4-chord. Let us investigate the intervals

between its notes {C,Eb,G,Bb}.

Table 2.3: Intervals within the C Minor 7th chord {C,Eb,G,Bb}

NOTE PAIRINGS SEMITONES (Ascending) SEMITONES (Descending)C and Eb 3 9C and G 5 7C and Bb 10 2 (C+ down to Bb)Eb and G 4 8Eb and Bb 7 5G and Bb 4 8

The first three distinct notes comprise the C Minor Triad {C,Eb,G}. From the table, we see

that these three notes are mutually consonant, since any pairing among them would only include

consonant intervals, both ascending and descending. However, upon the addition of the note Bb,

the mutual harmony is put into disarray. Granted, Bb is consonant to both Eb and G, but with C

(where it is 10 semitones above C, and 2 semitones below C octave) it is dissonant. This illustrates

the argument by Cocos and Fowers (2011).

The knowledge of harmonic chords will come in handy when we discuss the graph representations

of scales and modes in the next chapter.

2.3 Scales

Scales are basically extensions of chords in that they are both characterized in the same way, only

that, in general, a scale is longer and so is more tightly-spaced (denser) than a chord (since chords

and scales both partition the same “string” of 12 semitones). In a way, a scale is similar to Zn in


Algebra, in that n (the “octave note”) is fundamentally equivalent to the 0 element (the “tonic”

note) under the operation modulo n.

For instance, the C Major Scale {C,D,E, F,G,A,B}, which is in nature a Diatonic Scale, (and

we shall define this in a short while) repeats itself – but only an octave higher – after the 7th element.

Hence, the C Major Scale (or any Diatonic Scale, in fact) is comparable to Z7, with C as the 1st

element (or 0-element), ... , B as the 7th element (or 6 ∈ Z7), and C+ as the (7+1)th = 8th element

(which, just like 7 in Z7, is no longer written as part of the set); C+ ≡ C(mod7) as to 7 ≡ 0(mod7).

And this is precisely a clear illustration of the Law of Octaves.

The term Diatonic Scale is an umbrella term that refers to all 7-distinct-note scales that follow

the Law of Octaves. Common examples include the families of the Major and Minor Scales. We

should remark, however, that the Minor Scale family itself is composed of three members or sub-

families: Natural Minor (N. Minor), Harmonic Minor (H. Minor), and Melodic Minor (M. Minor).

All three serve specific purposes in the overall harmony of the composition. Moreover, even the

Major Scale family itself has seven sub-families too. These are the Modern Greek Modes, among

which the original Major Scale itself (Ionian) which we already know, and the Natural Minor Scale

(Aeolian). We shall go into more detail about modes in the next section.

Now we shall characterize scales according to their respective Interval Sequences, just as we did

with chords. For reference, we shall adhere to our examples in C.

2.3.1 Major and Minor Scales

In this chapter, we shall take the Major Scale as our reference scale (however, in the next chapter

we will argue that we can in fact discuss these concepts arbitrarily, which is actually one of our

main goals). That is, here in Music Theory we shall derive all other scales from the nomenclature

of the Major Scale. Loosely speaking (since we will be writing this in a more rigorous form in the

next chapter), we can simply write an arbitrary Major Scale (in any key or tonic) as the sequence

{1, 2, 3, 4, 5, 6, 7} where “1” refers to the tonic note; “2” the Major Second ; “3” the Major Third ;

“4” the Perfect Fourth; “5” the Perfect Fifth; “6” the Major Sixth; “7” the Major Seventh, all

with respect to the tonic. Writing a “b” before a number means it is a flat note, while writing a

“#” before a number means it is a sharp note. Other scales will be written with respect to these

assignment of notes in the Major Scale.

Having established that notation, we shall write the three Minor Scales in that form as well.

The following table summarizes the Interval Sequences of scales, including their nomenclature with

respect to the Major Scale.

And just as we did with p-chords, we will denote a k-scale by a sequence of k elements (excluding

the octave note), with a corresponding Interval Sequence of k elements consisting of W’s, H’s, and,


possibly, linear combinations of them, whose total is 12.

Table 2.4: Major and Minor Scales (in C)

SCALE NOTES INTERVALS SCALE IN CMajor {1, 2, 3, 4, 5, 6, 7} {W, W, H, W, W, W, H} {C, D, E, F , G, A, B}N. Minor {1, 2, b3, 4, 5, b6, b7} {W, H, W, W, H, W, W} {C, D, Eb, F , G, Ab, Bb}H. Minor {1, 2, b3, 4, 5, b6, 7} {W, H, W, W, H, H+W, H} {C, D, Eb, F , G, Ab, B}M. Minor {1, 2, b3, 4, 5, 6, 7} {W, H, W, W, W, W, H} {C, D, Eb, F , G, A, B}

The indicated Melodic Minor (M. Minor) sequence is only for the ascending pattern; descending,

it reverts to the Natural Minor (N. Minor).

2.3.2 Other Scales in Various Cultures

• Pentatonic Scales

In various parts of the world, the Pentatonic Scales (5 distinct notes) are commonly used

in traditional folk music. They were employed by the people of Greece, Scandinavia,

Eastern Europe, The Caribbean, West Africa, East and Southeast Asia, among others,

in the music of their tribes. They were also evident in some of the compositions of

Debussy and Chopin. One of the most common examples of the presence of Pentatonic

Scales is in Chinese melodies.

The Major Pentatonic Scale is {1, 2, 3, 5, 6}; whereas, the Minor Pentatonic Scale is

{1, b3, 4, 5, b7}. The C Major Pentatonic Scale is given by {C,D,E,G,A}, and the C

Minor Pentatonic Scale by {C,Eb, F,G,Bb}.

• Blues Scale

The Blues Scale is an American invention in the tradition of Blues, a precursor of Jazz.

Blues implies sorrow, longing, despair, and sometimes a call for help. Its conception

is attributed to Black Gospel, a music sub-genre centered about underground Chris-

tian worship of the African-American slaves of the late 19th century. Eventually Blues

developed to become incorporated in the various forms of entertainment and personal

expression, often performed in intimate gatherings.

The Blues Scale is a Hexatonic Scale (6 distinct notes) formed by {1, b3, 4, b5, 5, b7}.The C Blues Scale is given by {C,Eb, F,Gb,G,Bb}.Note that some literature list two types of Blues Scale: Major Blues and Minor Blues.

The Major Blues is given by {1, 2, b3, 3, 5, 6}. Since the Minor Blues is more prominent

in music in general, we shall limit our discussion to it, and refer to it as The Blues Scale.


Moreover, notice the uncanny resemblance between the Blues and the Minor Pentatonic.

The former is just the latter plus “blue note” (or flattened note, which gives it an added

morose feeling), the Gb (or b5). The intention is to make a gradual, draggy climb from

F to G, implying a heavy feeling.

The following table summarizes the nomenclature and interval sequences of the Blues

and Pentatonic Scales, including their examples in C.

Table 2.5: Pentatonic and Blues Scales (in C)

SCALE NOTES INTERVALS SCALE IN CMajor Pentatonic {1, 2, 3, 5, 6} {W, W, H+W, W, W+H} {C,D,E,G,A}Minor Pentatonic {1, b3, 4, 5, b7} {W+H, W, W, W+H, W} {C,Eb, F,G,Bb}Blues {1, b3, 4, b5, 5, b7} {W+H, W, H, H, W+H, W} {C,Eb, F,Gb,G,Bb}

• Scales Across Cultures

Different cultures have come up with their own scales, and of course, their own musical

instruments. Which is why when you hear an Indian chant you might hear a totally

different sound than what you hear from a classical Western performance. This is because

they have come up with a different system of discretization of notes - or in some cultures

maybe they did not even discretize their notes at all, but they treated notes as if in a

continuum because, strictly speaking, notes are in a spectrum and not in a roster. In

that sense they might be more correct than the Western perspectives in Music. However,

such spectral treatment of notes would entail a much more complicated system of Music

Theory (such as in Indian modes, or Ragas).

So here we shall present some scales, mostly from the Middle East and Far East, which

have been approximated (in sound) with respect to the Major Scale. If you play them

on a Western musical instrument (such as the piano, or flute) you will get a feel of what

they sound like, but only just as close to the genuine experience if it were played using

the musical instrument of that culture, and by a member of that culture itself.

Table 2.6: Other Culture-Specific Scales (in C)

SCALE NOTES INTERVALS SCALE IN CJapanese Insen {1, b2, 4, 5, b7} {H, 2W, W, W+H, W} {C,Db, F,G,Bb}Rock Pentatonic {1, b3, 4, b6, b7} {W+H, W, W+H, W, W} {C,Eb, F,Ab,Bb}Neutral Pentatonic {1, 2, 4, 5, b7} {W, W+H, W, W+H, W} {C,D, F,G,Bb}Scottish Pentatonic {1, 2, 4, 5, 6} {W, W+H, W, W, W+H} {C,D, F,G,A}Spanish Gypsy {1, b2, 3, 4, 5, b6, b7} {H, H+W, H, W, H, W, W} {C,Db,E, F,G,Ab,Bb}


As we have previously noted, there could be a myriad of scales if we consider each subset of

the Chromatic Scale. For the purpose of discussion, we shall focus on the scales mentioned above,

together with the Modern Greek Modes which we will be discussing next. A good knowledge of how

these scales are formed – both in the musical and mathematical sense – are adequate to help us

understand other forms of scales as well.

We should also point out that although there are 12 distinct notes in a Chromatic Scale, we could

generalize our discussion by categorizing scales into families, making the tonic arbitrary. In that

way we would not have to discuss a particular scale in all its 12 possible keys. This is valid since,

as we saw earlier, a scale is characterized by its own sequence of intervals, which stays uniform per

family of scale (e.g., Major Scale Family, Natural Minor Scale Family, Blues Scale Family, et al.),

regardless of the chosen tonic (which bears the name of the scale). Later we will no longer have to

indicate the key to a scale (e.g., we will no longer distinguish between C Major and D Major Scales

et al., but rather just call them collectively The Major Mode) once we treat chords and scales as

arbitrary mathematical objects.

And therefore, this leads us to the discussion on Modes.

2.4 The Seven Modern Greek Modes of Music (18th century

onwards)

Since the time of Pythagoras, sequences of notes have been altered to suit a particular “mood”,

or more appropriately, a “mode”. Mode in Music Theory, loosely speaking, refers to a particular

arrangement or “permutation” of the interval sequence of the Major Scale in the purpose of varying

the “mood” of the music.

One example would be the Church Mode (an Ancient Greek Mode) used by Greeks in wor-

ship chants. This was employed to make a distinction between church music and music performed

elsewhere (in royal courts, in public squares, in private chambers, et al.). In antiquity, Greeks de-

vised their own modes, but in changing times, some of their original modes were altered to suit the

prevailing taste and needs of each period.

To date there are seven Modern Greek Modes. Some of them were actually part of the original

scheme of Greek modes, some of them are altered versions of the original, and some of them are even

completely new modes. They are called (in order): Ionian, Dorian, Phrygian, Lydian, Mixolydian,

Aeolian, and Locrian. They were ordered in the way they permute the Major Scale, from the 1st up

to the 7th note. Each mode is built by setting the corresponding note of the Major Scale the tonic

and proceeding in the usual circular pattern. Hence, they simply permute the order of the five W’s

(whole tones) and two H’s (half tones or semitones) in the Major Scale. And this permutation is


what varies the “mood” and “flavor” of the music.

As a brief illustration in C, the C Ionian Mode is just the unaltered, non-permuted C Ma-

jor Scale {C,D,E, F,G,A,B}. If we start with the second note, it becomes the D Dorian Mode

{D,E, F,G,A,B,C}. So that is how it works. Following that pattern we will have the 7th mode,

the B Locrian Mode, {B,C,D,E, F,G,A}.Before we further our discussion, it is important that we first clarify the distinction between a

mode and a scale. A mode simply refers to the arrangement of arbitrary whole tones and half tones

among adjacent notes. So when we talk about the arrangement of intervals like {W, W, H, W,

W, W, H}, we mean the “original” Major Mode, or the more aptly called Ionian Mode. But if we

materialize this arbitrariness and assign notes between which these intervals fit, then it becomes a

Major Scale – like C Major Scale {C,D,E, F,G,A,B} and F Major Scale {F,G,A,Bb,C,D,E} –

both of which conform to the ruling on interval sequence established by the Ionian Mode.

Hence, in a way, a mode is arbitrary, whereas a scale is particular, in the sense that the latter

has to indicate a specific tonic. A mode is a general rule for the permutation of W’s and H’s (or

W+W’s, W+H’s, and H+W’s) in a succession of notes, and a scale is a particular assignment of

named notes fitting in that rule.

Now let us give brief remarks about each of the seven Modern Greek Modes and their respective

“moods” and roles in music.

• Ionian Mode

The term Ionian was assigned by Heinrich Glarean (Henricus Glareanus) in 1547 while

working on a system of church modes, which were not all incorporated in the current

seven Modern Greek Modes. This mode is identical to the original Major Mode itself.

• Dorian Mode

The Dorian Mode is was one of the most prominent church modes during the Middle

Ages. It is also commonly used In Celtic and African music, in folk, and rock music.

Moreover, it has a special place in Jazz music, and is often called the Jazz Minor Mode.

One of the most notable and influential Jazz tunes of all time, So What by Miles Davis,

is in Dorian Mode, as well as the relatively recent, Herbie Hancock’s Maiden Voyage.

• Phrygian Mode

This mode has a noticeable Arabic/Spanish/Moorish flavor. It is colorful and melodic,

and is commonly used in rock guitar riffs. Being exotic in nature, it is a staple in Fla-

menco music, but could rarely be heard in classic Western tunes.


• Lydian Mode

The Lydian mode is quite versatile, in that it can be used in Classical, Jazz, Fusion, and

Rock Music. It is also widely used in Rock and Country music, being the preferred mode

by musicians (especially rock guitar leads) in their solo improvisations. In Jazz, horn

players John Coltrane, Miles Davis, Ornette Coleman, and Woody Shaw also used the

Lydian Mode. Another popular example of a tune in Lydian is the main theme from the

TV cartoon, The Simpsons.

• Mixolydian Mode

Perhaps the most preferred mode in Free Jazz improvisation, the Mixolydian Mode is

highly deviant from the Classical Western sound. It is characterized by a bluesy, jazzy

feel. Having that kind of flavor, the Mixolydian Mode is the preferred mode for Funk,

Jazz, and Blues music. Musicians who want to inject a jazzy or bluesy feel to their music

are noted for the heavy use of a combination of the Mixolydian and Dorian Modes. An

example of such experimentation is the iconic Norwegian Wood by The Beatles, which

is a very simple yet profound tune about a nostalgic affair, and which is remarkable for

the use of an Indian sitar.

• Aeolian Mode

The Aeolian Mode is simply the Natural Minor Mode. Needless to say, he list of tunes

that have used this mode is endless. In general, songs written in the Minor Mode (Ae-

olian) tend to feel heavier, draggier, darker and more morose than their Major Mode

(Ionian) counterparts. An example would be the classic tune, Summertime, by George

Gershwin.

• Locrian Mode

This is the most impractical and generally useless of all the Modern Greek Modes. Its

existence is often considered to be just enforced and theoretical, that it was only included

in the list in order to complete the full roster of seven diatonic modes. Having said that,

it is often dubbed as the Theoretical Mode, a dummy mode with no apparent practical

use. And truly, when you play it on an instrument, it sounds strangely and disturbingly

dissonant, and does not clearly fit in any existing music genre. It is absolutely frowned

upon in Classical Music because of its disconcerting instability.

The following table summarizes the seven Modern Greek Modes and illustrates their examples


when applied in C (and by then they become scales: C Ionian Scale, C Dorian Scale, C Phrygian

Scale, et al.). Again, note that they are just rearrangements of the original Major Mode {W, W,

H, W, W, W, H}, and each mode is obtained by starting with the second note of the previous and

then proceeding in the usual circular pattern.

Table 2.7: The Seven Modern Greek Modes (in C)

MODE NOTES INTERVALS SCALE IN CIonian {1, 2, 3, 4, 5, 6, 7} {W, W, H, W, W, W, H} {C,D,E, F,G,A,B}Dorian {1, 2, b3, 4, 5, 6, b7} {W, H, W, W, W, H, W} {C,D,Eb, F,G,A,Bb}Phrygian {1, b2, b3, 4, 5, b6, b7} {H, W, W, W, H, W, W} {C,Db,Eb, F,G,Ab,Bb}Lydian {1, 2, 3, #4, 5, 6, 7} {W, W, W, H, W, W, H} {C,D,E, F#, G,A,B}Mixolydian {1, 2, 3, 4, 5, 6, b7} {W, W, H, W, W, H, W} {C,D,E, F,G,A,Bb}Aeolian {1, 2, b3, 4, 5, b6, b7} {W, H, W, W, H, W, W} {C,D,Eb, F,G,Ab,Bb}Locrian {1, b2, b3, 4, b5, b6, b7} {H, W, W, H, W, W, W} {C,Db,Eb, F,Gb,Ab,Bb}

To further illustrate this scheme, let us apply the modes to various tonics to build scales. For

instance, in the F Ionian Scale {F,G,A,Bb,C,D,E}, we will have G Dorian, A Phrygian, Bb

Lydian, C Mixolydian, D Aeolian, and E Locrian, all of which contain exactly the same notes as the

F Ionian, only rearranged. And another example, the Ab Lydian tells us that Ab is the 4th note

in some Ionian Scale. Which means that such Ionian scale is no other than the Eb Ionian Scale

{Eb, F,G,Ab,Bb, C,D}.

Chapter 3

Chords, Scales, and Modes:

A Mathematical Perspective

This chapter is at the heart of our study, where we observe the underlying mathematical patterns

and symmetries in musical chords, scales, and modes. In this pursuit, we will explore the results by

Cocos and Fowers (2011).

At this point, it would be safe to assume that the reader has already aquired a fair understanding

of the musical concepts of chords, scales, and modes, as discussed in the previous chapter. Moreover,

it would be best to remember the significance of intervals bearing 3, 4, 5, 7, 8, or 9 semitones

(“consonant intervals”), regardless of the chosen tonic.

In this chapter, also, we will treat the above musical concepts as arbitrary mathematical objects;

that is, we will no longer have to refer to particular chords, scales, and modes (as in C Major Chord,

C Major Scale, C Ionian Mode, et al.).

Lastly, at the pinnacle of these discussions, we will use a little Graph Theory (network maps) to

illustrate patterns and inherent symmetries, as well as the connections between modes and harmonic

chords.

3.1 Cocos and Fowers (2011)

In 2011, Mikhail Cocos and Shawn Fowers pubished a paper, Music by Numbers, which aimed at “...

presenting a mathematical way of defining musical modes.” From our previous chapter, recall that a

“scale” is a set of specified notes following an arbitrary rule for the permutation of whole tones (W)

and half tones (H), which is called a “mode”.

The definitions and theorems we will be presenting in this section are revised versions of theirs.

Nonetheless, they still carry the original significance of their results.

23

Chapter 3. Chords, Scales, and Modes: A Mathematical Perspective 24

Definition 1 (k-MODE)

A k-mode is an increasing sequence {ni} , i = 1, . . . , k of k natural numbers, with tonic n1 := 1

(fixed) and 1 ≤ k, nk ≤ 12. The value of ni, i = 1, . . . , k, is equal to the number of semitones the ith

note is above the tonic, plus 1. For i = 1, . . . , k,

nk+i ≡ ni(mod12) (3.1)

The difference,

di = (ni+1 − ni) , i = 1, . . . , k (3.2)

which may or may not be constant, refers to the number of semitones between the ith and (i+1)th

note in the sequence; dk is defined to be the number of semitones between the last note in the sequence,

nk, and the octave note nk+1 := 13.

Moreover,

k∑i=1

di = 12 (3.3)

Definition 2 (k-SCALE)

A k-scale is a sequence of k notes, 1 ≤ k ≤ 12, with a specified tonic and excluding the octave

note, corresponding to the k terms of a k-mode.

Illustration:

The universal set for all k-modes is the full 12-mode – or the Chromatic Mode – given by

{ni = i} , i = 1, . . . , k

or simply the sequence {1, 2, . . . , 11, 12}Any chosen tonic will be assigned to n1 = 1, and since the Chromatic Mode is symmetric with

notes equally-spaced at 1 semitone apart, then each succeeding note will be assigned numbers 2,. . . ,

12, respectively. For instance, in the C Chromatic Scale (for simplicity let us use just flats)

{C,Db,D,Eb,E, F,Gb,G,Ab,A,Bb,B}C = 1, Db = 2, D = 3, . . . , Bb = 11, B = 12

C+ = 13 ≡ 1 = C(mod12)


or in the F Chromatic Scale {F,Gb,G,Ab,A,Bb,B,C,Db,D,Eb,E}F = 1, Gb = 2, G = 3, . . . , Eb = 11, E = 12

F+ = 13 ≡ 1 = F (mod12)

As a matter of fact, an octave note of any mode will always be assigned to a value 12n+ 1,

where n ∈ N. Note that any other k-mode is a subset (or subsequence) of the 12-mode, such as the

following:

Major Mode (7-mode): {1, 3, 5, 6, 8, 10, 12}Blues Mode (6-mode): {1, 4, 6, 7, 8, 11}

For instance, let us consider the corresponding 7-scales and 6-scales of the above modes for the

tonic D:

D Major Scale (7-scale): D = 1, E = 3, F# = 5, G = 6, A = 8, B = 10, C# = 12

D Blues Scale (6-scale): D = 1, F = 4, G = 6, Ab = 7, A = 8, C = 11

In other words, a k-mode is a numerical sequence while a k-scale is a literal sequence, and both

of them have k terms. Notice that the numerical assignment for each letter is not fixed (e.g., if C

is the tonic, then 6 = F ; but if D is the tonic, then 6 = G) but rather depends on the k-mode, the

corresponding k-scale, and the choice of tonic.

It goes without saying that any choice of tonic for a particular k-scale will correspond to some

arbitrary k-mode (whether musically pleasing or not). Hence, we could say that a single k-mode is

“mapped” to twelve particular k-scales (one for each possible choice of tonic, up to enharmonics).

We refer to these k-scales that correspond to the same k-mode as Equivalent k-scales.

For instance, the Major Mode (a 7-mode) is mapped to the twelve Major Scales: C Major, Db

Major, . . . , B Major, whose repective numerical assignments follow {1, 3, 5, 6, 8, 10, 12}.Hence, all Major Scales are equivalent; all Blues Scales are equivalent; and so on. And in truth,

they are musically equivalent; equivalent scales may be notches above or below one another, but

nevertheless they sound similar.


Referring to some of the scales we have discussed thus far, now we bundle equivalent scales into

modes:

• 5-modes

Major Pentatonic Mode, Minor Pentatonic Mode, Japanese Insen Mode, Rock Penta-

tonic (“Chinese Jue”) Mode, Neutral Pentatonic (“Egyptian”) Mode, Scottish Pentatonic

Mode

• 6-mode

Blues Mode

• 7-modes

the Seven Modern Greek Modes (Ionian (Original Major), Dorian, Phrygian, Lydian,

Mixolydian, Aeolian (Natural Minor), Locrian), Harmonic Minor and Melodic Minor

Modes, Spanish Gypsy (“Jewish”) Mode

Furthermore, their paper referred to the “consonant intervals” – those intervals of 3, 4, 5, 7, 8,

or 9 semitones – as harmonic intervals. These “pleasantly-sounding” intervals are used to build up

harmonic chords in which, as we previously defined, each interval is a harmonic interval.

For the purpose of systematic notation in the succeeding discussions, let us denote the set of all

possible number of semitones in a harmonic interval by H.

H := {3, 4, 5, 7, 8, 9}

In relation to a k-mode {ni}, harmonic intervals occur between any ith and jth notes for which

|ni − nj | ∈ H. The two notes need not be adjacent.

Definition 3 (p-CHORD)

A combination of notes, a p-chord {mj} , j = 1, . . . , p is a subsequence of a k-mode (2 ≤ p ≤ k),

wherein ∀j = 1, . . . , p, mj = ni for some i = 1, . . . , k.

Remark:

Unlike a k-mode, the first element of a p-chord need not be 1. Taking just any p elements from a

k-mode creates a p-chord. However, this means that these chosen elements could have gaps or skips

in between, as far as indices are concerned.

For instance, in a 7-mode {n1, . . . , n7}, one possible 3-chord would be {n2, n4, n6}. Mathemat-

ically, this looks quite unpleasant, with all its skipping indices. So by definition, we can simply

rewrite this as the 3-chord {m1,m2,m3}, with m1 = n2, m2 = n4, and m3 = n6.


Definition 4 (HARMONIC p-CHORD)

A harmonic p-chord is a p-chord {mj} , j = 1, . . . , p, wherein ∀k, l = 1, . . . , p,

|mk −ml| ∈ H := {3, 4, 5, 7, 8, 9}.

Remark:

In other words, a p-chord is harmonic iff the absolute value of each pairwise difference between any

of its two terms belongs to H.

For instance, {1, 5, 8} is a harmonic chord since |1−8| = 7, |1−5| = 4, and |5−8| = 3 all belong

to H. Whereas, {1, 5, 8, 10} is a non-harmonic chord since |8− 10| = 2 /∈ H.

Two significant results from their study were the following theorems (nonverbatim):

Theorem 1 If a p-chord is harmonic, then p ≤ 3.

Theorem 2 Two equivalent k-scales have the same number of harmonic p-chords.

To paraphrase Theorem 1, harmonic chords occur only either as “triads” (3-chords) or note

pairs (2-chords, which are obviously trivial). We have already clearly illustrated this using our

counterexample in Subsection 2.2.4. Next we shall provide proof for this statement.

And then the implication of Theorem 2 is quite obvious since equivalent k-scales correspond

to the same k-mode, which we know to have a certain number of harmonic chords.

The following are the proofs of the two theorems.

Proof. (Theorem 1) Let {mj} , j = 1, . . . , p, be a harmonic p-chord. Suppose p > 3. (Goal:

Find a contradiction.) Since p > 3, then m4 exists. Now consider the elements m1,m2,m3,m4.

Now we can simply get rid of the absolute value in |mk −ml| for k > l since {mj} is increasing.

Consider the telescopic sum (m4 − m3) + (m3 − m2) + (m2 − m1) = (m4 − m1) ∈ H. Looking

at the elements of H, the only possible value for (m4 −m1) would be 9; otherwise, it would occur

that (mk − ml) /∈ H for some 1 ≤ l < k ≤ 4. But if (m4 − m1) = 9, then it could only mean

that (m4 −m3) = (m3 −m2) = (m2 −m1) = 3. Thus, (m3 −m1) = (m4 −m2) = 6 /∈ H, which

contradicts the assumption that the p-chord {mj} , j = 1, . . . , p, is harmonic. Therefore, p ≤ 3 (or,

“a harmonic chord can have up to three notes only.”) �


Illustration:

Major Mode: {1, 3, 5, 6, 8, 10, 12}C Major Scale: {C,D,E, F,G,A,B}

The following are harmonic chords from the above 7-mode (7-scale):

{1, 5, 10} = {C,E,A}

{5, 8, 12} = {E,G,B}

{3, 6, 10} = {D,F,A}

Whereas, the following chords are non-harmonic:

{1, 5, 10, 12} = {C,E,A,B}

{3, 5, 8, 12} = {D,E,G,B}

{1, 3, 6, 10} = {C,D, F,A}

Proof. (Theorem 2) Let S1 and S2 be equivalent k-scales. Although S1 and S2 have different tonics,

they still correspond to the same k-mode , and hence, the same numerical sequence representation.

It follows then that S1 and S2 have the same set of p-chords. Therefore, S1 and S2 have the same

number of p-chords. �

3.2 The Marriage of Music and Math:

Reconciling Differences

In Chapter 2 we characterized scales and modes with respect to the Major Scale and Major Mode.

As per Music Theory notations, we denoted the Major Scale by {1, 2, 3, 4, 5, 6, 7}, and the Major

Mode by the interval sequence {W, W, H, W, W, W, H}. In Music Theory, these “numbers” are

not really numbers, but merely symbols; no mathematical operation could be performed on them.

Which is why in some scales derived from the Major Scale, we used some non-mathematical symbols

like flat (b) and sharp (#).

Our goal in this section is to map these notations to the mathematical notations we just formu-

lated, using the definitions presented in the previous section. The juxtaposition of notations from

the two disciplines shall by no means give rise to a confusion or inconsistency in notation. Instead,


this shall set the marriage of the two of the most crude and natural disciplines – Music and Math.

After all, what are notations but merely abstract representations that aim to portray an underlying

greater meaning.

Now we can talk about the universal set for k-scales and k-modes, the Chromatic Scale or 12-

mode. Using Music Theory notations, we can build this up by “filling in the gaps” in the Major

Scale using flats, (for simplicity):

Major Scale: {1, 2, 3, 4, 5, 6, 7}Chromatic Scale: {1, b2, 2, b3, 3, 4, b5, 5, b6, 6, b7, 7}

Take note of the following enharmonics: b2 = #1, b3 = #2, b5 = #4, b6 = #5, and b7 = #6

Thus we can create the following mappings. The first column lists the notations used in Music

Theory (with reference to the Major Scale, as most literature do); the second columnn lists their

equivalent natural number representations in Math, without using ay kind of scale or mode as

reference. This gets rid of the need to rely on the Major Scale for notations.

Table 3.1: Notations in Music Theory and in Math

MUSIC THEORY MATH e.g. Tonic C1 1 C#1/b2 2 C#/Db2 3 D#2/b3 4 D#/Eb3 5 E4 6 F#4/b5 7 F#/Gb5 8 G#5/b6 9 G#/Ab6 10 A#6/b7 11 A#/Bb7 12 B


We only used the tonic C for illustration. The tonic is, by default, assigned a mathematical

notation of 1; each succeeding note is then assigned an integral value from 2 to 12, depending on

its “distance” (i.e., number of semitones) from tonic. In Music Theory, we assigned a 1 also for the

octave note; equivalently in Math we will assign a 13 ≡ 1(mod12) for the octave note. And so, the

“dispute” has been resolved.

3.3 Patterns and Symmetries

Having defined, mathematically, the essential musical concepts (k-mode, k-scale, p-chord, harmonic

p-chord) and using the two theorems, we can now explore the inherent symmetries and patterns

possessed by them. For visual representation we will employ some elements of Graph Theory (paths

and nodes) to create networks of notes.

Since equivalent k-scales are – to use the term loosely – “isomorphic” and hence fall under the

same “family” or k-mode, then we can simply describe one such family (k-mode) using a single

network (since we have shown that the choice of tonic does not matter, at least mathematically). In

the graphs or networks that we will be creating, we will see the harmonic p-chords.

Before sketching these networks, let us organize the information we have thus far collected about

some 5-, 6-, 7-, and 12-modes. We will list only the mathematical notations for the k-modes and the

accompanying interval sequences (just refer to Chapter 2 for their respective equivalent notations in

Music Theory).

It would be necessary to recall the concept of the difference between adjacent notes di =

(ni+1 − ni), for i = 1, . . . , k, which gives the interval sequence of W’s, H’s and some combina-

tions of them. In Math, this translates to W = 2, H = 1, and W+H = 3 = H+W. For instance,

the interval sequence {W, W, W+H, H, W, W} translates mathematically to the interval sequence

{di} = {2, 2, 3, 1, 2, 2}, for i = 1, . . . , 6, characterizing some 6-mode. Note that the sum of the terms

in the interval sequence is always 12, regardless of the kind of k-mode, since it is just a partitioning

of the “string” of length 12 semitones.

The following figures represent the modes we have discussed. Each figure, or network, represents

a k-mode, encompasing all 12 possible k-scales corresponding to it. For instance, the network for

the Major Mode applies to the C, G, Eb, Ab, D, et al. Major Scales, since they are all equivalent

scales belonging to the Major Mode.

Each network is labelled clockwise, in such a way that the top node (the one in the “north”

position) is n1 := 1. The nodes of each network are precisely the elements of the k-mode {ni}, as

described in the above table.

And as for the paths, two nodes ni and nj are connected iff |ni − nj | ∈ H := 3, 4, 5, 7, 8, 9.


Table 3.2: Mathematical Notations for k-modes

MODE NAME K-MODE {ni} INTERVAL SEQUENCE {di}Major Pentatonic {1, 3, 5, 8, 10} {2, 2, 3, 2, 3}Minor Pentatonic {1, 4, 6, 8, 11} {3, 2, 2, 3, 2}Neutral Pentatonic {1, 3, 6, 8, 11} {2, 3, 2, 3, 2}Rock Pentatonic {1, 4, 6, 9, 11} {3, 2, 3, 2, 2}Scottish Pentatonic {1, 3, 6, 8, 10} {2, 3, 2, 2, 3}Japanese In Sen {1, 2, 6, 8, 11} {1, 4, 2, 3, 2}Blues {1, 4, 6, 7, 8, 11} {3, 2, 1, 1, 3, 2}Spanish Gypsy {1, 2, 5, 6, 8, 9, 11} {1, 3, 1, 2, 1, 2, 2}Harmonic Minor {1, 3, 4, 6, 8, 9, 12} {2, 1, 2, 2, 1, 3, 1}Melodic Minor {1, 3, 4, 6, 8, 10, 12} {2, 1, 2, 2, 2, 2, 1}Ionian (Major) {1, 3, 5, 6, 8, 10, 12} {2, 2, 1, 2, 2, 2, 1}Dorian {1, 3, 4, 6, 8, 10, 11} {2, 1, 2, 2, 2, 1, 2}Phrygian {1, 2, 4, 6, 8, 9, 11} {1, 2, 2, 2, 1, 2, 2}Lydian {1, 3, 5, 7, 8, 10, 12} {2, 2, 2, 1, 2, 2, 1}Mixolydian {1, 3, 5, 6, 8, 10, 11} {2, 2, 1, 2, 2, 1, 2}Aeolian (Natural Minor) {1, 3, 4, 6, 8, 9, 11} {2, 1, 2, 2, 1, 2, 2}Locrian {1, 2, 4, 6, 7, 9, 11} {1, 2, 2, 1, 2, 2, 2}Chromatic {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}

Lastly – and most importantly – as a result of Theorem 1, the harmonic 3-chords in a k-mode

are the triangles formed by any three vertices in its network repreentation.

The first figure we will present is the universal mode, the Chromatic Mode or the 12-Mode. Any

possible mode is contained within it. Each of the other networks are contained within the network

for the Chromatic Mode.


Figure 3.1: The Chromatic Mode


Major Pentatonic Minor Pentatonic

Egyptian (Neutral Pentatonic) Chinese Jue (Rock Pentatonic)

Scottish Pentatonic Japanese Insen


Blues Spanish Gypsy

Harmonic Minor Melodic Minor


Ionian Dorian Phrygian

Lydian Mixolydian Aeolian

Locrian

Chapter 4

Conclusion

This brief research on the inherent mathematical patterns in musical chords, scales, and modes

focused primarily on defining these concepts in terms of mathematical structures. Our main goal

was to standardize the notations for chords, scales, and modes into forms using only the first twelve

positive integers, corresponding to the twelve standard notes of Western music (up to enharmonics).

And this notation scheme is completely arbitrary and independent the key of the scale. That is,

the standardization of their numerical representations made it possible to obtain a single form for

all equiavalent chord, scale, and mode types, regardless of the chosen tonic or starting note (which

dictates the scale key and chord name).

We anchored most of our discussion to an earlier research by Cocos and Fowers (2011). Their

paper, Music By Numbers, focused on depicting some common scales and modes, specifically the

standard Major Mode and the Japanese Insen Mode, by graphs or networks showing paths connecting

consonant notes, or notes that sound good together. One major difference between their study and

ours, however, was that in defining the Chromatic Mode, they fixed A to be the first element, and thus

A := 1 for all chords, scales and modes. It followed then that A#/Bb = 2, . . . , G = 11, G#/Ab = 12.

It is highly probable that they did this following the alphabetical order of notes,

Their interesting idea and initiative served as our model in attempting to standardize notations,

without fixing any note to be the first in the sequence. In our study, we declared that ANY of

the twelve distinct notes could be a starting note, and thus could be assigned to the numerical

value 1. And then from there the succeeding chromatic notes will be assigned the values 2, . . . , 12,

respectively. For instance, we can have the ssignment C = 1, C#/Db = 2, . . . , A#/Bb = 11, B = 12,

or F = 1, F#/Gb = 2, . . . , D#/Eb = 11, E = 12, and so on. By default, the octave note (the tonic

note in the next identical cycle of notes), was assigned a numerical value of 13. The Chromatic

Scale/Mode was likened to Z12, with tonic at 0 and the octave note at 12. Similarly, all the other

k-scales and k-modes are akin to Zk, with the octave note at k.

Perhaps one of the major points raised in this study was the difference between a mode and a

scale. We hope we have made it absolutely clear to the readers that a Mode is an arbitrary rule

for the spacing of notes, described either by numerical assignment to notes, or by the corresponding

Interval Sequence whose elements are the number of semitones in each space between arbitrary

notes. On the other hand, a Scale the resulting sequence of notes (as opposed to positive integers

36

Chapter 4. Conclusion 37

in a Mode) upon choosing a desired tonic, such as our examples above with the C and F Chromatic

Scales. A Mode is a fixed numerical sequence of some or all of the first twelve positive integers,

while a Scale is a varying sequence depending on the tonic (but whose spacing between elements is

still invariant).

What this important delineation tells us is that in Music, Relative Pitch is just as important as

Absolute Pitch. The former tells us the relative “distance” or “spacing” between notes, regardless

of which notes they are exactly; the latter tells us the actual sound of each specific note. While

Absolute Pitch is essential for musicians, the real understanding of Music comes from understanding

what Relative Pitch is all about. What charecterize a piece of music are not necessarily the notes

being played, but the relative distnce between those notes. That is why key changes (traspositions)

are often employed in playing written music without altering the feel or vibe of the piece sigificantly,

for it still preserves the relative distances of notes. And to put things into perspective, in terms of

the concepts we have tackled we could say that a Mode is to Relative Pitch, as a Scale is to Absolute

Pitch.

Now the inspiration for our network models came from Cocos and Fowers (2011), albeit altered

significantly since they fixed A := 1 and we did not (instead, we made them arbitrary). Their earlier

study dealt with a 5-Mode (Japanese Insen), a 7-Mode (Major or Ionian), and the full 12-Mode

(Chromatic). In our study we tried to diversify and included a few special scales and modes from

some world cultures such as the various kinds of Pentatonics or 5-Modes (Major, Minor, Neutral

(Egyptian), Rock (Chinese Jue), and Scottish), the iconic Blues (a 6-Mode), the moorish Spanish

Gypsy (a 7-Mode) or “Arabic/Jewish/Israeli” with a hint of dark Middle Eastern twang, the other

two Minor Modes (Harmonic and Melodic, which are also 7-Modes), and the seven phases or Modern

Greek Modes of the Major Scale.

In each network, we connected pairs of terms (represented by integers) whose absolute difference

is either 3, 4, 5, 7, 8, 9, which correspond to the standard Western notion of “pleasent sound”. These

intervals, respectively, are the Minor/Major Thirds, Perfect Fourth, Perfect Fifth, and Minor/Major

Sixths, which are universally accepted (or at least in the Western perspective) as the intervals that

produce harmony within an octave. While the inherent perceived beauty of Perfect Fourths, Perfect

Fifths, and Octaves was originally an idea of the Pythagorean School, progressing taste in modern

music has also included the Thirds and Sixths in the harmonic category.

Notice that the networks for the 5-Modes we presented in the previous chapter – with the

exception of the Japanese Insen – are simply central rotations of one another by an angle2π

5rad

or 72◦. Similarly, those for the Seven Major Greek Modes are also central rotations of one another

by an angle2π

7rad. This just goes to show that a little rearrangement or permutation of spacing

between arbitrary notes could add a different flavor or character to the music. Some modes, though,

Chapter 4. Conclusion 38

have no apparent symmetrical relation to the others, such as the Blues, Spanish Gypsy, Harmonic

and Melodic Minors, and a myriad others not tackled here. Nonetheless, it cannot be contested that

any conceivable mode is but a subset of the Chromatic Mode, and its network representation could

be derived by simple picking out portions of the Chromatic Mode network.

And one brilliant result and effect of these networks is that we could conveniently find the har-

monic 3-chords in them. The harmonic 3-chords in each mode could be traced by simply highlighting

the triangles (if any) formed by any three nodes.

Before we finally wrap things up, it would be best to recollect that all these ideas came from

the mere curiosity and obsession of Pythagoras with patterns, especially the telescopic pattern of

the fractions1

2,

2

3, and

3

4corresponding to the ubiquitous intervals of Octaves, Perfect Fifths, and

Perfect Fourths. It is quite impressive that these seemingly trivial patterns in the size of strings,

anvils, or any vibrating matter and a scholarly obsession have escalated to great proportions and

led to the foundations of Western Music Theory.

All these things only stand to give us insights on the innate similarities and links between the

disciplines of Music and Math, which gained prominence at the dawn of modern times, but could

actually be as old as time itself.

List of References

[1] M. Cocos and S. Fowers, Music by Numbers, Cornell University Library, New York, 2011.

[2] R. Hall, The Sound of Numbers: A Tour of Mathematical Music Theory, Saint Joseph’s Uni-

versity, Philadelphia, 2008.

[3] J. Hammond, Mathematics of Music (undergraduate paper), University of Wisconsin - La Crosse,

Wisconsin, 2011.

[4] A. Papadopoulos, Mathematics and Group Theory in Music, Cornell University Library, New

York, 2014.

[5] M. Richards, Pythagoras and Music, Rosicrucian Digest, California, 2009.

[6] S. Shah, An Exploration of the Relationship Between Mathematics and Music, University of

Manchester, England, 2010.

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