Music Theory - Introduction

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Introduction to Music Theory

Here I'll give a brief introduction to music theory.  Music is a combination of melody and rhythm. In this page, I'll be dealing mostly with melody.  So the rhythm aspect of music is kept in the background.

Note that the definitions of various terms may not be exact.  I have defined the terms in the sense in which they will be used in this page.

Notes

A note stands for a single pitch, or a single uniform sound associated with  a particular frequency. Any melody consists of a sequence of notes.

I don't know if you've heard Julie Andrews sing "Doe a deer" in the movie "Sound of Music".  If you have, you'll remember her singing "Do Re Mi Fa So La Ti Do".  This could serve as a beginner's introduction to the concept of notes.  Each syllable ("Do", "Re", etc.) is sung as a note. (You may ask how come the first "Do" and the last one are the same note, though the two "sound" different...we'll come to this later).

In Western music, we have twelve different kinds of notes, lying in sequence in terms of pitch.  A note is said to be higher than (or "to the right of") another, if it has a higher pitch than the other. Similarly a note is said lower than (or "to the left of") another, if it's pitch is lower than the other.

Seven of these are called "natural" notes, and are represented by letters of the English alphabet from "A" through "G".  Thus the note "A" is lower than the note "B". The next higher natural note to "G" is "A", and this "A" is much higher than the previous "A" to the left of "G" (separated by F, E, D and C).  The five remaining notes are called "sharps" or "flats", and lie in between the natural notes.  A sharp or flat note is represented in one of two ways:

Using the name of the natural note immediately to the left of the note, followed by a "#" symbol (pronounced "sharp")

Using the name of the natural note immediately to the right of the note, followed by a "b" symbol (pronounced "flat")

Wait a minute, if the sharps or flats fit in between natural notes, there should be seven sharps/flats, since there are seven places to fit them. The answer is that there is no sharp or flat note between "E" and "F", and between "B" and "C".  Thus the twelve notes, in ascending order of pitch are:

A
A# (or Bb)
B
C
C# (or Db)
D
D# (or Eb)
E
F
F# (or Gb)
G
G# (or Ab)
... and A again

The notes continue in a cycle, where the note immediately next to G# is A.

Hold on...how can "A" be next to "G#", if "A" was way down before "A#" ? The answer is that any note is called by the same name as the note that is twelve notes higher to it.  However, the two notes are said to differ by an octave. Thus, in the above listing of notes, the "A" which comes after G# is said to be an octave higher than the starting "A".

What's so special about octaves? Any sequence of notes will sound similar to (but not the same as) the same sequence of notes played in a different octave.

Notes and their frequencies

Any note, given its octave, is directly associated with a frequency. I say "associated with", because a note is not the same as a single frequency, but is a collection of frequencies out which one of the frequencies turns out to have the strongest influence on how the note "sounds". Whenever I refer to the  "frequency of a note", I'm referring to this "most influential frequency" of the note.

Till now, we were talking about notes with higher pitches, etc.  Higher notes simply mean notes with higher frequencies. For example, in the sequence {"A", "A#", "B"}, "A" has the lowest frequency, and "B" has the highest.

The exact frequency of each note in western music is dependent on two facts:

The frequency of any note is exactly twice that of the same note when sounded an octave lower.

The ratio of frequencies between successive notes is a constant.

Making use of the two rules above, we find that by multiplying a frequency by a constant ratio twelve times (to go an octave higher), we are effectively multiplying that frequency by two. Thus,

(constant ratio) ^ 12 = 2

or

constant ratio = 2 ^ (1/12) or the twelfth root of 2

Thus, the frequency f (n+k) of a note that is k notes higher than a note with frequency f(n) is given by

f(n+k) = f(n)* ( (twelfth root of 2) ^ k)

The constant ratio mentioned above is also called a semitone.  Thus we say that "A#" is one semitone higher than "A", but one semitone lower than "B".

The frequencies of notes ranging from "C" of one octave to "C" of the next octave is listed below.
 
 

 
Observe that we started with 128 Hertz and ended with twice that value.

Scales

In simple tunes, which we hum all the time, all twelve notes may not play a part.  Even in complex tunes, some notes will usually occur much more frequently than others.  This gives rise to the concept of a scale of notes, which is nothing but a subset of the twelve notes. A note is on a given scale if it that note is present in the scale's subset of notes.

Consider a hypothetical scale containing the following notes:

 C E G A C (while representing a scale, we normally go in ascending order of ptich, and add the last note, C in this case, as the starting note in the next octave for continuity)

The note "A" is on the scale while the note "B" is not on the scale.

The starting note of a scale (C in the above example) is said to be the root note of that scale.

Common scales

It is possible to construct a large number of scales using different combinations of the twelve notes.  However, it turns out that some scales are more common than others.  One of these "common" scales is called the "major" scale, which consists of seven notes.  Once we have decided on a root note, we can find out the notes in the major scale for that root note in the following manner:

Say, we choose "C" as our root note.

Go 2 semitones higher (D in our example)

Go 2 semitones higher (E)

Go 1 semitone higher (F)

Go 2 semitones higher (G)

Go 2 semitones higher (A)

Go 2 semitones higher (B)

For the closing note (starting note one octave higher), go 1 semitone higher (C)

Thus the scale of C major is {C,D,E,F,G,A,B,C}.

Similar steps can be followed to find the major scale for any root note.  As a thumb rule, you can remember the formula {root note, +2, +2, +1, +2, +2, +2, +1}, or, to put it more simply, {2, 2, 1, 2, 2, 2, 1}.

The scale of D major turns out to be {D, E, F#, G, A, B, C#, D}. Observe that C major is the only major scale that contains only natural notes.

Some other common scales and their corresponding formulae are listed below:
 
 

Relative minor and major

Observe that both C major and A natural minor consist of the same set of notes (except, of course that the root notes are different). This forms a relation between the notes C and A. C is called A's relative major, and A is called C's relative minor.  Such relations exist for every note. The relative major of a note can be found by adding 3 semitones to that note, and the relative minor can be found by subtracting 3 semitones (or adding 9 semitones, which boils down to the same thing).

Notes and Scales in Indian and Western music

In Western music, successive notes are equally separated (by equal frequency ratios, as discussed earlier).  In Indian music this is not so.  However, the notes in Indian music can be sufficiently approximated to fit into the set of twelve notes used in Western music.

The notes in Indian music are denoted by the seven notes:
Sa Ri Ga Ma Pa Dha Ni Sa

The above set of seven notes are called the Saptaswara (Sapta means "seven" and swara means "sounds").

Here, I will not go into details of  the theory of Indian music. But a few similarities between Indian and Western music deserve mention here:

Barring Sa and Pa, the rest of the above notes in Indian music can correspond to different pitches in different cases.

The concept of root note (as in the root note of a scale, described before) in Western music relates directly to the concept of a shruti in Indian classical music.

The concept of a scale in Western music is similar to the concept of a raga in Indian classical music.

Given a note from the saptaswara, the corresponding note in Western music can be found only after knowing the shruti and the raga.

Now, we'll go back to our discussion of Western music.  To go to the next section, click here