The Arithmetic of Music by Nestor S. Pareja

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Arithmetic of



The Arithmetic of Music. Copyright by Dr. Nestor S. Pareja. Parts of this book may be reproduced solely for personal use. For information, telephone and fax no. (632) – 552-7911; mobile phone no. 09162571055; address 30-G RPR I, Padre Faura, Ermita, MetroManila, Philippines;




This booklet is a synthesis of information gathered from more than 30 years of experience in learning to play musical instruments without much help from formal training. Of the informal training received from some “naturally talented” musicians, curiosity in how they seem to have learned and retained knowledge easily spurred me to study the dynamics of music and translate these to practical ways that could be used by people like me who had the interest to play music but lacked the “natural talent” for it. Data were culled from available printed articles, books, encyclopedias on music and from informal talks with musicians, professional and otherwise and sporadic playing of various musical instruments.

Inasmuch as writing this book was not the original intent of the author, references were not documented but nevertheless gathered from reliable printed sources. This is a revised edition of the first that never got to be published because of time and financial constraints, as the author would like to believe. As the information and ideas were interesting and very clear, the translation and transfer did not seem to be a problem at the time of first writing. Copies of the original manuscript were shared with knowledgeable


people assumed to be interested. There was no response from most. Of the few who did, it was most probably out of kindness and was only meant to be polite. This was however taken by the author as an affirmation. Upon review of the original manuscript which has been kept in hibernation for several years, the truth behind the earlier response was revealed. The author himself discovered that contrary to the intent of simplifying the presentation, the medium used could barely be understood.

The author strongly believes that these ideas are useful and worth sharing. Revisions have been undertaken to make the book simpler and easier to understand. Principles of tone/sound production, its physical attributes and appreciation by the receiver have been integrated to add to an easier and better understanding of music.

The Author


The Arithmetic of Music

Table of Contents

Introduction 1

Chapter One: Music, Sounds and Silences 11

Chapter Two: Music Intervals 21

Chapter Three: Reading and Writing 43

Chapter Four: Musical Scales

An Overview 55

Chapter Five: Diatonic Scale, Major and Minor 65

Chapter Six: Modal Scales 75

Chapter Seven: Pentatonic and the

Whole Tone Scale 83

Chapter Eight: Music Chords 89

Chapter Nine: Melody, Musical Form

and Design 119



Music may be defined as the art and science of transmitting emotions or ideas with the use of sounds (of varying pitches) and silences that are considered pleasing to the listener.

Musical sounds or notes are produced by strings, tubes or percussed suitable materials.

A musical note is defined or characterized by its pitch (high, low or anything in between). The pitch of a note is determined by the frequency with which the string or the column of air in a tube vibrates. Together with the pitch, a musical note is also defined by its timbre. The timbre allows us to differentiate the sound of a guitar from that of a violin.

The timbre is determined by final sound a musical instrument produces. The final sound is influenced by the interplay of the primary tone and its overtones.

Let us take for example a guitar string vibrating at a frequency of 264 cycles per second. This string does not only vibrate at 264 cps on its whole length ( called the primary length. It also vibrates 528 cps (264 X 2) on ½ of its length. 528 cps is an overtone of 264 cps (the primary tone of the example guitar string).

This string also vibrates at 792 cps (264 X 3) on !/3 of its length. 792 cps is also an overtone of 264 cps.

This string vibrates at 1056 cps (264 X 4) on ¼, 1320 cps (264 X 5) on 1/5 its length and so on down the line.

These simultaneous vibrations influence how the final sound of that string will be produced by tha particular string. That final sound is the timbre.



By Dr. Nestor S. Pareja INTRODUCTION

Music is primarily an art.

This booklet deals with another side: the measurable, easily identifiable and easy to grasp but vital feature of music. This is neither the mathematics nor the physics of music. Rather this is reading, writing and the Arithmetic of

Music for those who appreciate music but feel a need to

understand how it works. Understanding how something works can help us better appreciate, further explore and put to use the potentials of many of its wonders. For those who can hum a tune, can sing a song but have confined these activities where others may not hear, applying the simple guidelines this booklet offers will increase their love for and involvement in music. It can open new doors to a bigger world of music.

Lord William Thomson Kelvin, an English physicist, said that one’s knowledge of a thing is meager and

unsatisfactory if that thing can not be expressed in numbers and can not be measured. This booklet


subscribes to this idea. This will introduce numbers that will allow us to measure and to describe music understandably and easily. This knowledge will enable us to read and write music sheets/scores and begin to play our favorite music instrument or yet be able to compose songs.

This is primarily for those with little or no background in music but who are willing to learn the essentials to understand how it works. These essentials will help us understand music articles, many of which presume such knowledge. This knowledge will make us understand what Key of C or C Scale and the other Keys or Scales mean. This will make us understand why a symphony or a song is a B flat minor or a D flat composition. We will understand what

Ionian Mode, Dorian Mode or Lydian Mode mean.

This is also for those who already are into music, the professionals, those with advanced knowledge and the “gifted”. This book presents a simple complementary view that will help those with advanced knowledge in music further grasp/comprehend/internalize and make the most of their present concepts and know-how.

Music teachers may find practical ways of teaching the basics of music.



The first two chapters deal with simple principles that are applied in music. They are explained even the “non-gifted” in the Sciences can understand. Waves, overtones and intervals are explained in layman language.

The third chapter introduces the language of music, symbols used by musicians to convey emotions and ideas, the same way we use letters of the alphabet, punctuation marks, words, sentences and paragraphs to transmit ours in the print media.

A complementary view to the present concepts of music is offered in the fourth chapter. For the novice this can advantageously be the primary view because it is simple and easy to use. For the advanced this view hopes to significantly strengthen their concepts, abilities and know-how.

The fifth to seventh chapters talk about music scales: how the different scales are created, why they project certain moods/ambiance/atmosphere and how and when they are used presently.

The eighth chapter discusses music chords: their uses, how they are built, and a simple and easy method of


The ninth chapter touches briefly on melody, musical

form and design. This will introduce terms that will help us

better understand and appreciate music.


The basic strategy of this book is repetition, saying the

same idea in different ways. The ideas are presented from different points of view, so that we may acquire a 3 (even a 4?) dimensional picture of the concepts. This is to ensure that concepts are well understood. They are important building blocks in our understanding of music. As is true in many disciplines, concepts in music are simple when taken individually/separately but seem complicated when taken all together at one time.

Chapters are arranged mainly to facilitate explanation. They are intended to stand alone and could be understood even without having full comprehension of the others. Although ideas are interrelated, a chapter may be understood without reading other chapters. Chapters generally refer to the same ideas but are taken from different views/perspectives.

Music concepts are basically simple but are prone to many different and complex interpretations, because of


mysteries/confusions created by vague statements. The use of numbers avoided such statements and has facilitated definition and explanation of concepts. Sentences in this book are meant to be simple and direct but inevitably repetitious because of the basic strategy.

Learning these concepts and keeping them at our fingertips will allow us to devise our own learning

exercises for our chosen musical instruments and/or music activities. This will allow us to determine our own pace and our own level of involvement. We will not be

limited by availability of specific learning materials because hopefully we will be able to devise our own. We are offered choices.

Numbers are liberally used so ideas can be measured, can be easily described or defined and stated in no uncertain way.

The importance of practice and of guidance by a competent music teacher can not be overemphasized.

When we students are ready, the teacher in this book will appear.



Music is the rhythmic, melodic, harmonious and

colorful arrangement of sounds and silences that communicates emotions and/or ideas.

Frequencies and timbres distinguish musical sounds

and tones.

Differences in timbre are due to differing overtones present. The timbre allows us to distinguish the sound of a violin from that of a trumpet or the voice of a friend from another’s. The timbre makes us like or be pleased with certain sounds. As beauty is in the eyes of the beholder, how pleasing and how harmonious sounds are is personal and relative.

Recurrence and interval are the measuring sticks or parameters used to assess how pleasing sounds are. In this book it is assumed that sounds which are heard more often are more pleasing.

Interval in music is the ratio or relation of the frequency

of a note to the frequency of another. Dividing the frequency of a note with the frequency of another will


give us that ratio or interval. In Western music two kinds of intervals are identified. One measures the interval created by

notes with a common reference note, the tonic. The other

measures the interval between nearby or adjacent notes. In the first kind, the most pleasing interval is the

unison. It is created by notes with the same frequency as

the tonic. Almost as pleasing are those created with the

octaves (the 8th natural note or semitone 13). The

octaves are notes whose frequencies are either ½ or 2

times the frequency of the tonic.

The next pleasing intervals, after the unison and the octaves, in descending order, are created by notes whose frequencies are 3 times, 4 times, 5 times, 6 times and so on, that of the tonic. These frequencies are correspondingly produced by the (1/2), 1/3, ¼, 1/5, 1/6 and so on the other

secondary lengths of a vibrating string. The shorter the

secondary length, the higher the frequency of the note/sound the vibrating string produces and the note produced is less harmonious with the tonic.

The frequencies of the notes within an octave (a group of 8 natural notes) are partials of the frequencies produced by the secondary lengths. This is explained further from another view and in a more easily digestible form in Chapter



The second kind of interval is that created between adjacent notes. There are 2 types that are identified, the

semitone (or half tone, half step, a 16/17 or its reciprocal

17/16, interval) and the whole tone (or whole step, a 16/18 or its reciprocal 18/16, interval). If we divide the frequency of one note with the frequency of the next higher adjacent/nearby note and get the ratio 16/17 or 0.94, the interval is a semitone. If the ratio between the frequency of the lower note with the frequency of the higher adjacent note is 16/18 or 0.89, the interval is a whole tone.

In the chromatic scale there is only one kind of interval, the semitone. Dividing the frequency of the lower semitone with the frequency of the higher gives us the ratio of a semitone interval. The inverse/reciprocal (frequency of the higher divided by the frequency of the lower note), likewise defines a semitone interval.

Musical scales with 2 types of intervals between adjacent natural notes, whole tones and half tones, are called diatonic scales.

The notes of a melody may sound lacking in fluidity, rhythm and unity if played without other sounds. The notes of music chords correct these. They supply additional


“color” and character to the melody. They serve as the framework and body to a set of melodic notes.

Silences between notes play a significant role as well. Tryreadingasentencewithoutspacesinbetweenwords.

The character, personality, atmosphere, ambiance or mood of a set of melodic and harmonic notes depends on the location of the semitone and whole tone intervals. Thus we have the emergence of the different scales: the diatonic, the modal, the pentatonic, the whole tone and the major and the minor scales. These scales on analysis are derivatives and/or modifications of the mother scale, the chromatic


Dr. Nestor S. Pareja Author




Music is the arrangement of sounds and silences to express emotions and/or ideas.

This is achieved through the use of rhythm,

melody, harmony and “color” of sounds. Rhythm is the

uniform or patterned recurrence of a beat, accent, or the like. Melody is how the musical notes of varying frequencies or pitches are arranged one after another while harmony is how pleasing two or more musical notes sound together. “Color” is the quality, the timbre, the mood, the atmosphere, the personality or the character of the melody or harmony of sounds/notes. “Color” is determined by what overtones are present. Overtones will be explained later.


Sounds are waves that are heard. Other waves are

seen. Others are felt. The ripples created by a stone

thrown into a quiet body of water are waves.

Waves that are seen are those coming from a light bulb and those that are reflected/bounced back from objects. Reflected light waves give color to the objects. Objects that bounce off all light waves of natural light will look white. Objects that do not bounce off light waves will look black.

Light waves are energy and energy is heat. Dark colored clothes do not reflect most of the light waves they receive. Most are absorbed. Light colored clothes bouncing off most of the light waves they receive explains why they feel cooler than are dark colored ones.

Ultraviolet waves are felt. These waves can warm or burn. Taste buds in the mouth and nerve endings in the nose possibly respond to waves coming/emitted from what is tasted or smelled. The frequency of the wave determines whether it will be heard, seen, felt, tasted (?) or smelt (?).

The function of the receiving sense organ also determines how a stimulus will be interpreted or translated.


To illustrate; the function of the eye is to see light. Stimulation of the eye, other than by light, will still be interpreted as light. As an example, touch your eye with your eyelid closed. The eyelid will feel your finger. A dark spot with a halo of light about the area of the finger that is in contact with the eyelid will be sensed or “seen”.


A wave is a rhythmic/regular/patterned movement of energy.

A guitar string when plucked will vibrate and produce waves. We hear some of these waves. The other waves we feel as we hold the guitar close to our chest.

Imagine and trace the up and down movements of a point, the midpoint of the whole length of the vibrating string. The whole length of the string is called its primary length. From position A, it moves up to position B, bounces down to position C (passing through position A) and bounces back to position A, completing a cycle. The string completes many more cycles with diminishing loudness until the energy transferred to it by plucking is used up.


If the up and down movement of this point is plotted on paper that is behind and moving horizontally (across) at an even speed, the vertical distance AB (also equal to AC) traveled by the midpoint on the paper is called the

amplitude of the wave. The amplitude of a wave determines

its loudness, volume or intensity. The horizontal distance traveled by the midpoint on the paper during one cycle is called the length of the wave or wavelength. The number of cycles completed during/within a unit of time is called the

frequency of the wave, example, cycles per second (cps or Hertz, Hz). The frequency determines the pitch/key of a

note, the higher the frequency, the higher the pitch/key. Joseph Fourier, a French physicist, discovered that any complex wave could be broken down into its


component simple sine waves. Waves are simple or complex. A sine wave is the simplest waveform. It has a constant amplitude and frequency.

He discovered that a complex wave like the sound wave produced by a performing orchestra is but a combination of many simple sine waves of different amplitudes and frequencies produced by the different musical instruments.

Have you wondered how the sounds of all the music instruments are heard from a single speaker, further considering that all those sounds are picked up from a single record disc by only one laser beam or one phonographic needle?

The complex wave on the disc is picked up by the laser beam/needle and transmitted to the speaker. The speaker reproduces the complex wave. The ear “breaks” the complex wave into its recognizable component simpler


waves allowing us to recognize the sound of each instrument.

How then do we distinguish the sound of a clarinet from that of an oboe? They are distinguished by the

difference in the shape of the wave each produces.

The shape of the wave depends on what overtones are present and predominate.



A vibrating guitar string produces a complex wave. It does not produce only one sound wave. Sound waves are simultaneously produced by the whole length, by the ½ length, by the 1/3, by the ¼ length, by the 1/5, 1/6, 1/7 and on the other secondary lengths of the string. They have different frequencies.

Let us take for example, a guitar string 1-meter long (its primary length), vibrating at 264 Hertz (its primary

tone) on its primary length. At ½ meter, this string is

also vibrating at 528 Hertz which is 2x the frequency of the primary tone.

The ½, 1/3, ¼, 1/5, 1/6 meter and so on are called the secondary lengths of the 1-meter primary length. On its 1/3 meter, it will be vibrating 3x that of the

primary tone, 3x 264 =792; on the ¼ meter, 4x 264 =

1056; on the 1/5 meter, 5x 264 = 1320, and so on. These statements say that the frequency of the wave

produced on a secondary length is equal to the frequency of the primary tone multiplied by the reciprocal or inverse of the secondary length. The

reciprocal or inverse of ½ is 2/1 or 2; that of 1/3 is 3/1 or 3; ¼ is 4 and so on.


On the other hand, the volume of the sound produced diminishes in direct proportion with the secondary length. The volume of the wave produced by the ½ secondary length is ½ that of the primary length. The volume and intensity of a wave is determined by the wave’s amplitude. The amplitude is directly proportional to the secondary length of a vibrating string.

Each of the secondary lengths is producing overtones of their own on so-called tertiary lengths. Likewise these tertiary lengths are producing overtones of their own on the quaternary lengths and so on down the line.

The overtones produced on the secondary

lengths are called harmonic overtones: on the ½, the 2nd harmonic overtone, on the 1/3, the 3rd harmonic

overtone, on the ¼, the 4th harmonic overtone and so

on down the line.

In summary, a vibrating string produces a sound, which is the result of the combination of its

primary tone, harmonic and other overtones. These

different waves influence and/or interfere with each other before producing the final waveform. This combination


of overtones and the resulting final wave form will give this particular vibrating string its timbre.

When a string vibrates, it makes surrounding objects vibrate/resonate on their natural frequencies. The string also initiates forced frequencies on these objects. The presence of two or more waves or the so-called superposition of waves creates interference (reshaping of the wave). The compactness/density and shape of the vibrating material, force in stretching or the tension or pressure applied, surrounding atmosphere, humidity and many other factors determine what overtones will prevail and thus define the final shape of the wave produced, its timbre, its “color”.

“Color” of Sounds

For some people, sounds of musical instruments are easily distinguishable from one another. It is not that easy for others. This is due to varying levels of sensitivity to small differences in the shapes of the wave. This

ability to distinguish subtle differences in the “color” of a sound can be acquired and developed.


revealing scientific experiment is worth mentioning. A group of kittens was raised in an environment whose lines were mainly horizontal and another group in an environment whose lines were mainly vertical.

When they matured and were released to normal environments, the “horizontal group” kept running into legs of chairs, tables and objects whose lines were mainly vertical They “could not see” vertical lines. Those in the vertical group would bump on object whose lines were mainly horizontal.

On microscopic examination, differences in their brain cell structures were identified and authenticated, verified and/or validated.

This experiment demonstrates that light waves influence brain cell development. Light and sound are waves. Waves share common properties. It can be inferred that sound waves influence brain structure as well.


CHAPTER TWO Music Intervals

We learned about overtones, how they affect the shape of the final wave, the “color” of sound. We learned

how resonance, interference and other factors affect the shape of the final wave. When a string vibrates, it does not produce only one wave. It produces waves of different frequencies on its primary, on its secondary, tertiary and other lengths. The waves/overtones produced on the secondary lengths are specifically called harmonic

overtones. Overtones produced on the lengths other than

the secondary lengths are referred to simply as overtones. Both the secondary and the other overtones are among the factors that influence the final shape of the wave. The waves initiated from among surrounding materials, their interaction and superposition also modify the shape of the final wave. The resulting wave give the sound produced by the vibrating string or any vibrating material its timbre or “color”.

The next concept is music interval. A good grasp of these concepts is important in our study. They are basic blocks on which we will build a clear understanding of how music works and how we can make this knowledge work


with us.

Music interval is the ratio of the frequency of a

note with the frequency of another note. Ratio is the

proportional relation of one frequency to another. Dividing

the frequency of a note with that of another will give us

this ratio/interval. Ex: The frequency of note C is 264 Hz. The frequency of note G is 396. The interval between these notes can be stated either as 264/396 or its inverse 396/264 (2/3 or 3/2, if 264 and 396 are divided by 132, a number common to both, the greatest common number/divisor).

Music intervals are defined 2 ways: (1) intervals produced by notes with a common reference note, the

tonic and (2) intervals produced between adjacent/nearby

notes. The consecutive natural notes of the Key of C are given letter names: C, D, E, F, G, A, B, C’. Examples of the first kind of interval are those between notes C and D, notes

C and E, notes C and F, notes C and G, notes C and A,

notes C and B and finally notes C and C’. Note C is the

reference note, the tonic note of the Key of C.

Examples of intervals between adjacent notes are, those between notes C and D, notes D and E, notes E and F, notes F and G, notes G and A, notes A and B and finally notes B and C’. The interval C and D belongs to both kinds


of intervals.

The interval between adjacent notes in a diatonic scale is either a whole tone or a semitone (half tone or half step). The whole tone interval is practically or essentially equal to 16/18 (or 0.89), if we divide the frequency of the lower note with the frequency of the higher note. If we divide the frequency of the higher note with that of the lower note we get the reciprocal 18/16 (or 1.125).

The semitone interval is essentially equal to 16/17 (or 0.94) or its reciprocal 17/16 (or 1.0625).

In the popular Key of C whose notes are assigned letter names C, D, E, F, G, A, B, C’, note D is the 2nd natural

note and its interval with the tonic C is called a 2nd interval.

Note E is the 3rd natural note and its interval with C is called

the 3rd interval. The interval with the 4th natural note is

called the 4th interval and so on. There are other special

names for all these intervals as we shall see later. They are more descriptive and specific but somehow tend to muddle and confuse the issue for others. These names will however be discussed later for whatever purpose they may serve.

The most popular musical scale is the major diatonic

scale. This scale is associated with the familiar do-re-mi


also called Key of C, Scale of C or the C Scale. It has 7 easily distinguishable and convenient to remember tones or notes that are assigned the letters C, D, E, F, G, A, B. For purposes of practice and convenience these letter notes may be sung do-re- mi- fa- sol- la- ti.

If you tried to sing that, how did the scale end? Did it feel that something more was to come after the note -ti? The feeling that the –ti sound gave, describes what a leading

note does.

Next, instead of singing the scale ending in -ti, sing do- re- mi- fa- sol- la- ti- do. Doing so creates an atmosphere of finality.

These feelings are evoked, elicited or produced if one is familiar with the do-re-mi sound.

The C major diatonic scale is usually written C, D, E,

F, G, A, B, C’. Notice the prime sign (’) on the 2nd C. The

interval C to C’ is an octave. The term octave may also

refer to the 8th natural note of a scale, in this case, note C’.



The frequency of tonic C (264 Hz) divided by that of C’ (528 Hz) is ½. The inverse ratio, the frequency of C’ divided by that of C is equal to 2/1 or 2. These intervals are called

octaves. The note that creates an octave interval with the

tonic is also called an octave. The octave note is also called the 8th natural note or the 8th degree. In this book, every

now and then, we will refer to the octave note and the octave interval as semitone 13. The natural notes of a Scale are also known as degrees, 1st, 2nd, 3rd and so on.

The frequency of note C’ is equal to the frequency of the 2nd harmonic overtone of note C. The 2nd harmonic overtone is produced by the ½ length (one of the

secondary lengths) of a vibrating string and its frequency is

2x that of note C. Notes C and C’ being octaves produce essentially the same overtones; their timbres and “colors”


are essentially the same. Notes D and D’ are essentially the same notes, as they produce essentially the same overtones; but D’ belongs to the next higher octave, scale or key. The same is true with the other octave intervals (semitone13).

The interval between the tonic note C and the 2nd

natural note D is called a 2nd interval or a Major 2nd or M2 interval.

The interval between the tonic C and the 3rd natural

note E is called a 3rd interval or a Major 3rd or M3.

The interval between the tonic C and the 4th and the

5th natural notes are respectively referred to as Perfect 4

or P4 and Perfect 5th or P5.

The interval between the tonic and the 6th natural note is called a Major 6th or M6. The interval with the 7th natural note is called a Major 7th or M7. After the octave are the 9th, 10th, 11th, 12th, 13th intervals.


The harmonic overtones produced on the secondary

lengths (½, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9 and so on) are

progressively and proportionately increasing in frequency.

Harmony with the tonic however proportionately diminishes with the increase in frequency.

The amplitudes of the waves proportionately

diminish with the secondary lengths. And because the

amplitude determines the volume of the overtones, they are progressively less heard.

Intervals with the tonic are classified into: (This

classification is arranged in descending harmony with the


Perfect Unison Octave Dominant Subdominant Major Mediant Submediant Supertonic Leading Note Accidentals Augmented Diminished Minor PERFECT INTERVALS

Unison and Octave

The most pleasing and harmonious intervals are those from a (1) unison, the superposition of waves whose frequencies are the same, and from (2) octaves, the superposition of waves of the tonic and of waves whose frequencies are its halves or doubles.


The simultaneous playing of notes whose frequencies are an octave apart is practically a unison. Notes that are an octave apart produce practically the same overtones. The prime sign (‘) after the letter note is used to signify a note which is one octave higher, like C’ is one octave higher than C. The notes C, D, E, F, G, A and B will sound essentially the same as the notes C’, D’, E’, F’ G’, A’ and B’ although they are one octave apart.

The overtone created on the ½ length, the 2nd

harmonic overtone creates the octave interval with the tonic.

As mentioned earlier, the term octave also refers to the 8th natural note, like C’ is the octave of C, and the other

way around. Octave is also conveniently used to mean a

group of 8 natural notes. The group of the notes namely C,

C#, D, D#, E, F, F#, G, G#, A, A#, B and C’ has 13 notes. Although it has 13 notes, this group of notes is still called an

octave, because it has only 8 designated natural notes.


The overtone that creates the next most pleasing interval with the tonic is the 3rd harmonic overtone. The 1/3

secondary length produces this. Its waves dominate the


them as to loudness and as to influence in determining the final waveform of the tonic. The 1/2 partial of this overtone is assigned to the dominant and 5th natural note of the major

diatonic scale of its tonic.

The frequencies of the notes within an octave are equal to the partials of the frequencies of the harmonic overtones of the tonic.

The range of frequencies of the notes in an octave starts with the frequency of the tonic and ends with the frequency of the 8th natural note (the octave note, semitone 13, the 2nd harmonic overtone or the overtone produced by the ½ secondary length). In our reference scale the Key of C, the frequencies of the notes start with 264 Hz (tonic C) and ends with 528 Hz (note C’, the 8th natural note, the octave note of C, semitone 13 or frequency of the 2nd harmonic overtone of tonic C).

The frequency of the 3rd harmonic overtone of note

C is 264 x 3 = 792 Hz. This is beyond the 264 to 528 Hz range. If we divide 792 by 2 we get the frequency 396 Hz. This frequency is assigned to note G, the designated 5th

natural and dominant note of this reference scale, the Key

of C. As we shall see farther, the frequencies of the other notes in this scale, natural and accidental, are portions/parts


of or partials of the frequencies of the harmonic overtones produced by the secondary lengths.

The interval that the dominant note G creates with its tonic (note C) is called a 5th interval because the assigned

note to it is the 5th natural note of the C Scale. This interval

is also known as Perfect 5th or P5. The interval is 8

semitones. The frequency of note G is 264 x 3/2 = 396



Another perfect interval with the tonic is the Perfect

4th (P4). The interval it creates with the tonic is less

harmonious than that created by P5. The interval is 6

semitones. Its frequency is a partial of the 4th harmonic

overtone. The 4th harmonic overtone is produced by the ¼ length of a vibrating string. !/3 of this frequency is assigned to the 4th natural note of the diatonic scale of its tonic.

The 4th harmonic overtone of tonic C is 264 x 4 = 1056 Hz. 1/3 of 1056 is 352 Hz. This frequency is assigned to note F, the designated 4th natural note of the Key of C, In

the Key of C, note F is a natural note below the dominant note G. Note F is less dominant to and less harmonious with tonic C. Note F is consequently called the subdominant of


the Key of C


The overtone that creates the next most pleasing interval with the tonic is the 5th harmonic overtone. The 1/5

length of a vibrating string produces it. The frequency of the 5th harmonic overtone of note C is 264 x 5 = 1320. The ¼ partial of this frequency is 330 Hz. This frequency falls right in the middle of the frequency of the tonic (264 Hz) and that of the 5th note (396 Hz). This note is consequently called the mediant of the Key of C. This note is designated as the

3rd natural note of the C major diatonic scale. The interval it

creates with the tonic is called a Major 3rd (M3) or 5

semitones. The mediant of the C Scale is note E. Its

frequency is 264 x 5/4 = 330 Hertz.


A closely related note to the mediant is the

submediant, because its frequency is also a partial of the

5th harmonic overtone. The submediant is less

harmonious to the tonic than the mediant is, thus the


secondary length of a vibrating string. The frequency of

the submediant is 1/3 of the 5th harmonic overtone. The

submediant of the C Scale is note A. The interval it

creates with the tonic is called a Major 6th (M6) or semitone 10. The frequency of note A is 264 x 5/3 = 440 Hertz.

Let us temporarily stray away from the present topic and go to a subject deemed important and interesting at this point. This information will allow us to use the easily available piano music sheets in playing the same melody, in the same Key, in our selected music instrument, even if the assigned key/pitch of that instrument is not the same as that of the piano. Musical instruments have different assigned/recognized pitches or keys.

In an international music convention, the frequency 440 Hz has been set as the standard frequency for note A of the middle octave on the piano keyboard. But even if this has been set as the international standard, still 264 Hz, the frequency of note C of the piano keyboard, is used as the reference frequency in identifying the pitch of other music instruments.

Almost all Western music instruments can produce the popular Do-Re-Mi sound together with its accidental


notes. The frequency of the sound Do on an Alto Saxophone differs from the frequency of the sound Do on a standard trumpet.

The frequency of the sound Do of an Alto Saxophone is 313.5 Hz. This is the same frequency as that of note Eb on the piano. The Alto Saxophone is accordingly called an Eb instrument. The frequency of the sound Do on a standard trumpet is 231 Hz, same as the frequency of the note Bb of the piano. The standard trumpet is accordingly called a Bb instrument.

In producing the Do-Re-Mi- sound, the fingering (which holes are close or open) is the same for all kinds of saxophones (bass, tenor, alto, soprano), clarinets, flutes and piccolos. This means that if one knows how to play the do-re-mi sound on one of the mentioned music instrument, he/she can play the do-re-mi sound with the same fingering on all the other instruments. They will however differ as to Key or pitch. The key/pitch is determined by frequency of the Do- sound of the instrument, its assigned tonic. If the frequency of the tonic is the same as that of the C of the piano, the pitch/key of that instrument is C. If it is equal to that of note D of the piano its pitch/key is D.


The tonic determines the pitch or key of a Key or Scale. We have the 12 Keys or Scales, C, C# (Db), D, D# (Eb), E, F, F# (Gb), G, G# (Ab), A, A# (Bb) and B depending on the tonic. The other Keys or Scales have the same letter names but they belong to lower or higher octaves. They are signified by the use of small letters or prime signs.

Going back to intervals:

The next most pleasing interval with the tonic is that produced by the 7th harmonic overtone. It is produced by

the 1/7 length a vibrating string. The ¼ partial of this frequency is not assigned to any natural note in its scale. It is assigned to an “accidental note”. Its frequency is a semitone lower than the 7th natural note (semitone 12) and

thus is known as the minor 7th (m7), semitone 11 or note

Bb. Because its frequency is also a semitone higher than the 6th natural note (semitone 10) it is known as Augmented

6th (A6), semitone 11 or note A#.

Note A# and note Bb although called by different names have the same frequencies. They are referred to as enharmonic equivalents. The same is true with C# and Db, D# and Eb, F# and Gb and lastly G# and Ab.

Historically C#, D#, F#, G# and A# had lower frequencies than their flat equivalents until it was agreed during an international music convention to adjust, modify or temper the frequencies of the notes of the music scale and


assign them equal frequencies and call them enharmonic equivalents. Present day music scales are tempered. Semitone intervals are not exactly 16/17 or 17/16 in the same way that whole tone intervals are not exactly 16/18 or 18/16. They are however essentially and practically 16/17 and 16/18 intervals.

A natural note that is lowered or made “flat” by a semitone is signified by the flat sign b after the letter symbol of the natural note. A natural note made higher or made “sharp” by a semitone is signified by the sharp sign # after the letter symbol of the natural note.

The frequency of this accidental note is 264 x 7/4 = 462. Please read further on Accidentals.

Interestingly, note A# or Bb, even if its only an

accidental note has a higher ranking in the hierarchy of

harmony than a Major 7th (M7) semitone 12 or note B, a

natural note. Historically, it took many years of exposure to

the M7 before listeners came to appreciate the sound of this interval. M7 is a characteristic interval/sound of Jazz music.


The next most pleasing interval is that with the 9th

harmonic overtone. This is produced by the 1/9 secondary length of a vibrating string. The 1/8 partial of this frequency


corresponds to the natural note, one note superior (above) the tonic; it is thus called the supertonic. The supertonic of the C Scale is note D. It creates a Major 2nd interval, M2

(semitone 3) with the tonic. The frequency of note D is 264 x 9/8 = 297 Hertz.

Leading Note

Among the natural notes of a major diatonic scale, the least pleasing harmony/interval is produced by the 15th

harmonic overtone. It is produced by the 1/15 secondary length of a vibrating string. The 1/8 partial of this frequency

corresponds to the 7th natural note and creates a major 7th interval (M7) with the tonic (semitone 12). This is note B. It creates the greatest “tension” with the tonic, and being so, must invariably resolve or lead to the tonic. This note is thus called the leading note. The frequency of note B is 264 x 15/8 = 495 Hertz.



Augmented Intervals

Perfect and major intervals are augmented when they are lengthened or increased by a semitone. Raising the higher note a semitone does this. An augmented note is signified by the sharp sign # after the letter note. A perfect 5th, semitone 8 (C to G,) is augmented by raising note G a semitone, Thus a C to G# is an Augmented 5th, A5, +5 or

semitone 9. Another example is, a Major 6th, semitone 10 (C to A) is augmented by raising the higher note A, a semitone. Thus a C to A# is an Augmented 6th, A6 or

semitone 11.


Lowering the higher note of a perfect interval by a semitone diminishes the interval. A P5, semitone 8 (ex. C to G) is diminished to C to Gb. A diminished note is signified by the flat sign b. Thus C to Gb is a diminished 5th, d5, –5

or semitone 7.

Minor Intervals

Lowering the higher note of a major interval by a semitone makes it minor. A Major 3rd (C to E), semitone 5, becomes a minor 3rd, semitone 4, when E is lowered to Eb. A Major 7th C to B, semitone 12, becomes a m7, semitone 11, when B is lowered to Bb.


In a diatonic scale, as the term implies, there are 2 types of intervals between adjacent notes, (1) a whole

tone/whole step (16/18 or 18/16) or (2) a semitone (16/17

or 17/16) interval. The location of the semitone interval characterizes/distinguishes one modal scale from another. We will go further into this in a subsequent chapter.

In the Key of C, the intervals CD, DE, FG, GA, and AB are whole tone or whole step intervals. There are semitones in between; C C# D, D D# E, F F# G, G G# A and A A# B.


The intervals EF and BC’ have no semitones in between. These intervals are called semitones (half tones, half steps, 16/17 or 17/16).

As will be mentioned repeatedly, the location of the semitone interval determines the mood/atmosphere of a scale.

An example is: In the Key of C, the semitone interval is between the 3rd and 4th and between the 7th and 8th natural notes as shown in the following illustration. (Alternatively, this scale can be viewed as being composed of 2 similar and smaller scales which are C, D, E, F and G, A, B, C’. Both are composed of 4 notes, 1st three of which are whole tones apart, the 3rd and 4th notes are only semitones apart.)

In the Key of C minor, the semitone interval between the 3rd and 4th natural is shifted to between the 2nd natural and the 3rd natural (3rd natural of the minor scale); the semitone interval between the 7th and 8th naturals is



The concept of changing moods/atmospheres through shifting of the location of the semitone intervals will be more apparent in Chapter 6, Modal Scales.



Reading and Writing (Musical Notation)

Music is the organization of sounds and silences to

convey an idea and/or emotion. Music is usually transmitted by voice, by music instruments or by any sound producing gadget/device. Another method of conveying/transmitting musical ideas is with visual symbols that can be sung or played in music instruments. Most musicians can “hear” the music notations they read. Many of them are taught to read music notes just like students of today are taught the alphabet phonetically.


As non-musicians use visual symbols to convey ideas with letters of the alphabet, punctuation marks, words, sentences and paragraphs, musicians use another set of visual symbols. Musical notation is the use of this set of symbols.

We will start with symbols that show the pitch of a note. Later we will learn the symbols that show the location and duration of sounds and silences.


Music symbols are written on a set of 5 horizontal lines called the staff. The lines are numbered 1 to 5 from the bottom.

Note symbols are written on the lines or in the spaces between. The vertical location of the notes on the staff shows their pitch. Notes whose pitches fall beyond the pitch range of the staff are written on or between ledger lines.


Most notes have 2 parts, a note head and a stem.


Notes are given letter names. Letters C, D, E, F, G, A and B. They are commonly used interchangeably with do-re mi-fa- sol la- ti- (do). As we shall see later, they are not the same.

The illustrated note that follows is G;

We call it G because the letter G is “sitting on” the 2nd line and the note is aligned with G. The second line is assigned to the note G. The G written at the beginning of the staff is called the G clef. The staff with the G clef is called the treble clef.


Musicians, artistic as they are, use a fancy version of the letter G.

Notice the G clef curling on the 2nd line. THE BASS CLEF

The F clef is made up of a curved line and 2 dots. The F clef is written “on the 4th line”, one dot above and the other below. Consequently the 4th line is called the F line. The staff with the F clef is the bass clef.

An important reference point is the middle C. This is the C that is in the middle of the piano keyboard. The middle C is written on a ledger line immediately below the treble clef or on a ledger line just above the bass clef.



treble and the bass clef are connected with a brace. This is called the grand staff.


The Key of C has 7 natural notes (8 including the octave note). There are “unnatural notes” called

accidentals. To designate these accidentals, 2 symbols are

used. In the staff, to raise a note a semitone (or a half step), the sharp sign # is written before a note symbol. When used with a letter note, the sign is written after.

To lower a note by a semitone the flat sign b is used. When these signs are written in the staff, just after the G or F clef, they apply to all the notes on those spaces or lines in all the measures.

When these signs are written within a measure or a

bar, they affect only the notes in that particular measure.

After the bar line, the notes revert back to their previous pitch.




We learned that the vertical location of notes on the staff shows their pitch. We will now study the symbols that show duration of notes and silences.

We indicate the pitch and duration of a musical sound with a note symbol. The rest symbols indicate the location and duration of silences.



Shorter duration of notes is indicated by stems, shading the oval and flags on the stem.

The half note is half the duration of the whole note.

The quarter note is half the duration of a half note.

The eighth note is half the duration of a quarter note.

As we add a flag to the stem we shorten the duration by half.


Notice that so far the durations of the notes are multiples of 2. If we want a note to last ¾ the duration of a whole note, we combine a half note with a quarter note by connecting them with a curved line called a tie.


The most frequent combinations are those in which a note is tied to another note ½ its value. A dot placed after a note head, is a more precise and concise symbol than a tie. A dot increases the duration by ½ its value.


The duration of silences is equally as important as the duration of the notes.


These pauses are indicated by symbols called rests. As silences are without pitch, rest symbols are assigned specific and constant locations on the staff.


The fermata sign over a note means that the note is held longer that what the value signifies. How much longer is left to the discretion of the conductor or of a solo performer.


We have just learned the different kinds of notes and symbols used to show the relative duration of notes and silences. We have learned that a half note is half as long as a whole note, a quarter note is half as long as a half note and so on down the line. We have not learned however, how long to hold or play a whole note.


A beat is a regular pulsation, like the beat of a normal heart. When we run or get excited, the heartbeat goes faster. The tempo is faster.

Sometimes at certain position or posture, we can hear our heartbeats go RUB-dub, RUB-dub, RUB-dub, the RUB part we may hear louder than the dub part. This organization of heartbeats into strong and weak beat pattern is called the



To determine how long to play each note, we have to know what kind of note is equal to one beat. If for example, we know that a quarter note equals one beat, then a half note will have two beats and a whole note, four beats.


Bar lines are used to separate groups of notes into beat patterns (meters). The bar lines divide the staff into segments called measures or bars.


The time signature indicates the meter of a music piece. It consists of 2 numbers, written after the clef sign just at the beginning of the first measure. The upper number tells the number of beats per measure and the bottom number tells what kind of notes receives one beat. A time signature


of ¾ means there are 3 quarter notes to a measure or its equivalence in notes and/or rests. It means that the quarter note receives a beat and there are 3 beats to a measure.


One beat may be broken into pulses. Like for example, the sound of a train may be heard as RACK-e-ty, RACK-e-ty, RACK-e-ty. The RACK sound represents the

beat, and the RACK, the –e and the –ty sounds are the


CHAPTER FOUR Musical Scales

An Overview

(Complementary Viewpoint)

Almost everything in this book revolves around the concepts this complementary viewpoint advocates. One of the concepts is: The interval between adjacent semitones is the same throughout the whole chromatic musical

scale. This interval or ratio is either 16/17 (0.94) when we

divide the frequency of the lower semitone with the frequency of the higher semitone or 17/16 (1.0625) if we divide the frequency of the higher semitone with that of the lower semitone.

In Music, interval is the ratio/number we get when we divide the frequency of a music note with that of another. To


demonstrate: The frequency of note C is 264 Hz. The frequency of the next/adjacent higher semitone, note C#, is 280.5 Hz. If we divide 264 by 280.5 we get the ratio/interval 16/17 or 0.94. The interval between C# (280.5 Hz) and the next higher semitone, note D (297 Hz) is also 16/17. The interval between D and D#, D# and E, E and F, F and F# and so on, are all 16/17.

Assigning numbers to the semitones will make understanding and definition of the concepts easier. The fact that the intervals between adjacent semitones is constant means that as long as we retain the semitone template,

pattern or sequence, the intervals between notes and the reference note, the tonic (note C in our example), is

maintained. It also means that retaining the semitone

template, maintains its mood/atmosphere/melody. The

resulting combination of notes will however belong to another key/pitch/octave.

Example: The semitone template of

Do-Re-Mi-Fa-Sol-La-Ti-Do sound in the Key of C is 1-3-5-6-8-10-12-13.

Semitone 1 is note C. Using the same template but with note D as semitone 1 will create the same Do-Re-Mi-Fa-Sol-La-Ti-Do sound/melody, but in the Key of D. The resulting melody sounds “the same” but higher in pitch/key or octave


(another group of 8 natural notes).

The use of the semitone template is applicable to music chords (major triads, 7th chords, extended chords, 11th, 13th), Modal Scales, diatonic scales (major and minor) and many others.

Back to Basics

Many are familiar with a musical jewelry box. Each time such a box is opened, it plays a melody. The melody/music is produced mechanically by pegs or pins on the surface of a rotating cylinder that beat/hit/strike the tuned teeth of a comb like steel plate.

Imagine a comb like steel plate.

The teeth are tuned and arranged like the notes of the Key of C on the piano keyboard. However, unlike the keys on the piano where the black keys are shorter,


teeth representing the black keys are of the same size and are aligned with those of the white keys. Let us assign letters to the teeth of the steel plate: C, C#, D, D#, E, F, F#, G, G#,

A, A#, B, C’. These are the same letter names assigned to

the notes/keys of the Key of C on the piano. Letters with the sharp sign (#) represent the black keys on the piano which are the accidental notes of the Key of C. The white keys are its natural notes.

Imagine also a "cylinder" with pegs or pins on its surface that can strike the teeth of the steel plate, one by one, from left to right.

On "rotation" of the cylinder, these pegs will produce the following series of musical notes, C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C’. The progression of the notes is only by a


semitone interval. Such progression is called chromatic.

A musical scale that progresses in semitone interval is called a chromatic scale. Let us assign numbers 1, 2, 3, 4, 5, 6,

7, 8, 9, 10, 11, 12, and 13 to the pegs.

Next, let us remove the pegs that strike the teeth hitting the black keys, leaving behind only pegs numbers 1,

3, 5, 6, 8, 10, 12 and 13 in place. These pegs will strike C, D, E, F, G, A, B, C’. These are the natural notes of a major diatonic scale, the Key of C. Key of C because the

reference/tonic note is C. Let us call this cylinder the major


If we slide this major diatonic template one tooth to the right, the pegs will strike notes C#, D#, F, F#, G#, A#, C’ and C’#.

These are the natural notes of another major

diatonic scale, the Key of C#. Notice that there are black

keys in this C# Scale. The black keys are accidental notes only as far as the C major diatonic scale on the piano is concerned. Among the 12 major diatonic scales only the Key of C has all its natural notes in the white keys of the piano.


Sliding the template another tooth to the right will align the numbers of the template to the natural notes of the Key

of D.

Identifying the natural notes of any major diatonic scale will then be easy, just as easy remembering the sequence 1, 3, 5, 6, 8, 10, 12, 13. Peg 1 aligns with the tonic and defines/names the Key.

This happens because even if the template is moved/transposed, the intervals of notes with the tonic are maintained/preserved. All resulting groups of notes can be sung Do- Re- Mi- Fa- Sol- La- Ti- Do and “sound the same”. They will only differ in pitch/key but they will produce/create the same mood/atmosphere or melody. We can see then how melodies/songs can easily be transposed from one Key to another to adapt to the note/frequency range of a music instrument or a singer.


Music sheets are easily available for the piano. Transposing melodies for any music instrument from music sheets for an instrument with a different pitch/key can then be done by anyone with this knowledge.

We are familiar how the major diatonic template was created and how it is used. A template for any combination or sequence of intervals can thus be easily made and put to use. Example, a major chord triad which is made up of the 1st, 3rd and 5th natural notes of a Key. In the Key of C, these notes are C, E and G. The semitone numbers of these notes are 1, 5 and 8. Because 1 defines the tonic we do not have to memorize this number. We need to memorize only nos. 5 and 8 to identify the notes of a major chord triad.


the note 1, F# as the note 2(F# is the 5th semitone from D) and A as note 3 (A is the 8th semitone from D).

Transposing or changing of pitch or key of a melody, music scale and chords to fit a musical instrument, will then be possible without consulting charts and the like.



Diatonic Scales, Major and Minor

A diatonic scale is a progression of musical notes, which has 2 kinds of intervals between adjacent natural notes. The interval between natural notes can either be (1) a

whole tone or (2) a semitone. The whole tone interval is

16/18 (or 0.89) or 18/16 (or 1.125). The semitone interval is 16/17 (or 0.94) or 17/16 (or 1.0625).

The Key of C has 7 natural notes, 8 if we include the octave (note C’).


Let us include the semitone numbers:

The Key of C is called a major diatonic scale because the interval between its 1snd 3rd natural notes is 5

semitones. A 5 semitone interval is also known as a Major

3rd or M3. The semitone template of a major diatonic

scale is 1, 3, 5, 6, 8, 10, 12 and 13.

A diatonic scale whose interval between its 1st and 3rd

natural notes is 4 semitones is called a minor diatonic





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