Music Theory - Chords

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Chords and Harmony

Now that we know what notes are and what they sound like, how would it be like to sound two notes together?  While we're at that, how about sounding three, or more, notes together?  Would it sound good, or would it sound like Cacophonix at work?  These are the questions I'll try to answer in the following sections, and the discussion takes us through the concept of chords and the related concept of harmony.

Sounding more than one note together

When two notes are sounded together, if it sounds good, the notes are said to be in harmony with each other. Now, what, in absolute terms, does "sounding good" mean? I'll answer that later, in the section about the mathematics behind harmony. Till then.......well, it just means "sounding good"!

If three or more notes are sounded together, and they still sound good, that set of notes is said to form a chord. In fact, a combination of notes that does not sound good can also be technically called a chord, but it won't be a nice chord, that's all!

Types of chords

We have already found out that there are different types of scales identified by the root note of the scale and the rule governing the sequence of notes that form the scale.  We also have different types of chords.

As in the case case of scales, a chord is also identified by its root note and a rule specifying the relation between the notes in that chord.  Even the names given to some types of chords are the same as the names given for scales (Major and Minor). The differences are :

In scales, we think about one note (of the scale) at a time; in chords we sound all the notes (in the chord) together.

The rules specifying the relation between the notes of a chord are different from those for scales.  For example, the rule specifying the set of notes in the scale of A Major is different from that specifying the set of notes in the chord of A Major (Obviously, it won't sound good if all the notes in a scale are sounded together!).

Let us take the case of a common chord type - the Major chord. To explain the rule for the Major chord, let us take any note as the root note, say "A".  Here come the rules:

A Major chord has 3 notes.

One of the notes is, if you haven't guessed it already, the root note itself.

To get the second note, you add 4 semitones (click here for a refresher about semitones) to the root note.  In our example, adding 4 semitones to "A", we get "C#".

To get the third note, you add 7 semitones to the root note.  Adding 7 semitones to "A", we get "E".

Thus we have found that the chord of A Major consists of the notes "A", "C#" and "E".  Now, it doesn't matter which octaves these notes are sounded in.  As long the notes "A", "C#" and "E" are sounded together in some octave or other, the resulting sound is called "A Major".  This "independence from octave" holds for any chord.

Summing up our rule for the Major chord, we can represent the set of notes as { (Root Note), (Root Note + 4), (Root Note + 7) }.  Putting it in a more compact form, we have {0, 4, 7}. The table below lists the rules for some common types of chords, along with examples for each type. Note that some chords have three notes, and some have four.
 
 

 
The list does not stop here.  But I have given the more common types of chords that you are likely to encounter. As an exercise, try constructing the notes of different chords (especially of types Major, Minor and 7th), and memorising them.  It'll certainly come in handy.

Related chords

The relations between chords are closely linked to scales.  That means that some chords fit better than others, when sounded during a song in a given scale. Before finding out why this is so, it is necessary to touch upon the concept of the possible scales for a chord.

Take a sample chord, say C Major.  It consists of three notes, namely, C, E and G. One can raise the question : "Which are the scales that include the notes C, E and G ?". Choosing from a list of all possible major and minor scales, we get the following scales that include the notes C, E and G:

C major

F major

G major

A minor (natural minor)

D minor (natural minor)

E minor (natural or harmonic minor)

Let us call the above list of scales the "possible scales for the chord of C Major". A chord is related to C Major if all notes in that chord are included in any one of above list of chords. Let me put the same idea in a more formal and general way:

A chord X is related to a chord Y if all the notes that are part of X are included in any one of the possible scales for chord Y.

In our example, i.e., that of the chord of C Major, we find that the following are some chords that are related to C Major (considering only the scale of C major and A minor):

F Major
G Major
G seventh
A minor
D minor
E minor
.....and so on !

While exploring chords occurring in a song, we frequently find that the chords are indeed related (using "related" as we defined it above). So the concept of "related chords" is an important thing to keep in mind.

The mathematics behind harmony

This section is intended for those who are comfortable with a bit of maths, and who get bugged with the concept of "harmony" or "why some notes sound better when played together than others".

I have mentioned earlier that a note is not just one frequency, but a collection of frequencies.This is something to do with the mode of vibration of the source of sound.

For discussion purposes, we can think of a note associated wth a frequency f, as a sound wave that is the sum of waves of frequencies f, 2f, 3f, 4f and so on. The "strongest" component (the wave with the highest amplitude) of the combined sound wave is the wave with frequency f;  the wave with frequency 2f has a lesser amplitude, and so on. As we go towards the higher frequency components, their influence on the sound of the note becomes more and more trivial.

It is also a known fact in physics that when two waves are superimposed on each other, the resulting effect is most pronounced if the two waves are of equal frequency, and less pronounced if the frequencies vary.

An obvious conclusion from the above fact is that if the same note is sounded from different sources (a combination of f, 2f, etc merged with another combination of f, 2f, etc), the sounds will harmonize into one single note. Another conclusion is that if a note (a combination of f, 2f, etc) is sounded together with the same note in the next octave, whose fundamental frequency or "most influential frequency" is double that of the first note (a combination of 2f, 4f, 6f, etc), the components 2f, 4f, 6f, etc of the two notes will merge with each other, creating an effect of harmony.

A"not-so-obvious" conclusion follows. What happens if we merge a note of fundamental frequency f with another note of fundamental frequency (3/2)f ? One note has frequency components f, 2f, 3f and so on, while the other has (3/2)f, 3f, (9/2)f and so on. Notice that components 3f, 6f, 9f and so on are present in both the notes and thus, the two notes promise to produce harmony.

Now, the quesion arises : "Given any anote, which is that note which has (3/2) times its fundamental frequency ?". In the section about notes and their frequencies, I have stated that the fundamental frequency of a note is "twelfth root of two" times the that of its immediate lower note. So, recalling some of logarithm theory, we can say that:

When we are multiplying the fundamental frequency of a note by a factor of k, we are effectively moving up by "12 times log₂ (k)" notes.

Substituting k=3/2 in the above equation, we find that by multiplying the fundamental frequency with 3/2, we have moved up 7.02 notes ! We can approximate this to 7 notes. Thus, a note with 3/2 times the fundamental frequency of "C" is that which is 7 notes higher than "C", namely, "G" !

It is well-known that C and G indeed sound good together !

Experimenting with different  integer ratios, we obtain the following results (note that the octave of the note is not very significant in this discussion, since shifting a note by an octave just means multiplying the fundamental frequency by a power of 2)::
 
 

 
Thus the chord of C Major (containing notes C, E and G) has notes of approximate fundamental frequency f, 5f/4 and 3f/2. This set of ratios holds good for any major chord.Also, in the above table, notice that the notes for many of the important chords related to the note C appear : "F" as in F Major, "A" as in A minor (the relative minor of C Major), D# as in C minor and "Bb" as in C7th.

Though the above table is given for note "C", it is a straightforward task to extrapolate the results for any note. For example, G is to C what A is to D and what G# is to C#, and so on.

I hope the above discussion has given an insight into the maths behind harmony in music. To go to the next chapter, click here.