It is widely known that many popular songs of our era are comprised of only four chords. And more often than not, the four chords used in a four chord song - are the I (one), IV (four), V(five) and vi (minor sixth) chord.
There is in fact a mathematical explanation for why these four chords seem to work well to create music that the majority of mainstream audiences seem to like.
It takes a little thought to grasp this concept from a mathematical perspective, so grab your beverage of choice, sit in a comfortable chair and put your thinking cap on. This is only for those truly interested in a mathematical explanation of why certain chords work well together in popular music.
Music is based on math because sound consists of waves that can be measured and quantified mathematically. Different sounds produce wave forms in different frequencies. A particular note will produce a particular and unique sound print based on how fast the waves move up and down which is measured as “frequency”.
The mathematical relation of these frequencies to one another, account for the fact that some sets or groups of notes are harmonious with one another (sound good together) while others sound dis-harmonious (don’t seem to go together). Another way this can be described is that notes that sound good together are consonant, and notes that don’t blend are dissonant.
When a composer of a popular song sets out to write an upbeat or up-tempo song, they will more than likely choose a major key – which tends to evoke a happier vibe (if you will).
When selecting the chords to be used , the safest ones he/she knows will always sound good, are the I chord (related to the root note of the key the song is in) the IV chord (based on the fourth note of the key) and the V chord (based on the fifth note of the key). We often refer to these "notes" as the first, fourth and fifth degree of the scale.
The I, IV and V (one, four, five) chords are also known as the tonic, subdominant and dominant chords of the scale. These chords are always a safe bet in any song in a major key, because they will harmonize well with any note in the key the song is in.
The next safest bet mathematically speaking, is probably the vi (sixth minor chord) which is why if the composer adds a fourth chord and only a fourth chord to the three major chords described above - it will often be the vi chord.
Now I will attempt to explain mathematically, why the foregoing is true.
A major chord is comprised of a root, a major 3rd (4 semitones or two whole steps above root) and a perfect fifth (7 semitones or 3 and one half steps above root). These notes blend well together because of the way the sonic frequencies merge together and complement one another verses clashing with one another. A chord can be formed using any note as a root note.
The chords available for any given key which will sound correct with that key based on music theory, are limited to the chords which can be formed using the notes in that key. Any given diatonic key will have only 7 notes that are in that key and these are the 7 notes we can use to form chords that go with that key.
Since our melody notes will be taken from the notes in the key we are composing in, it follows that the chords that will sound good with the notes we choose for the melody, should be comprised of the notes in the melody. Therefore the chords that will support any melody in a given key must be formed using the 7 notes in that particular key.
Using the 7 notes in a major key, limits which chords we can form and only gives us three options for major chords that can be formed using those 7 notes. And those 3 options for major chords will always end up being the I chord, IV chord and V chord (all major).
While the foregoing explains why you don't have a choice of which chords you can use without venturing outside they key - it falls short of explaining why the I, IV, and V chord sound good in a given key.
To understand this better, we must revisit the idea presented in part one, that suggests that certain notes blend well together because of the way the sonic frequencies merge together and complement one another. Our brains will instinctively have a desire to gravitate towards complimentary frequencies that will blend together to form pleasing sounds. The relationship between the sonic frequency of two notes is described in music theory as an "interval" which is how far apart the sonic frequencies are - commonly measured in what we call semitones (with one semitone being the smallest step in a Western Music chromatic scale).
The most congruent and consonant sounding intervals are the unison (same exact frequency or 1:1 ratio) and an octave (exactly double the frequency or a 2:1 ratio). It's easy to visualize how the sound waves will line up evenly and blend together if you have exactly 2 crests of one wave for every one crest of the second.
Besides the octave and unison, the next most consonant (harmonious) interval is the perfect fifth. This is because the ratio between the sonic frequency of two notes that form a perfect fifth (7 semitones apart) is 3:2. Because these two numbers are small, the crests of the sound waves will peak at the same place more often than they would if the ratio were say 15:8. So any two notes with an interval between them of a perfect fifth, will sound good together.
If we start with the note of a particular key (say C in C Major for example) we can get to a perfect fifth interval from there by going up 7 semitones which lands on G if we start with C. That happens to end up being the 5th note in a diatonic major scale. (1)C (2)D (3)E (4)F (5)G. We know, that the interval between C and G in the key of C major will result in two notes that blend together because they form a perfect fifth, and we know that these notes will sound good together whether they are played at the same time or successively.
So if we build a chord using the G note as the root of the chord (since G is on the other end of a perfect fifth interval from the home note of our key (C), and that chord is formed using only the notes in our key, then it makes sense that the chord (in this case G Major) naturally evolves from the tonic I chord - C Major (which uses our key's home note as it's root).
It's like we use the tonic chord and pivot to the G chord because the G note is a perfect fifth interval using C at the other end. So the chord using this note as the root (G major) evolves naturally from the tonic chord C (with C as the root). This is why the five chord is the dominant.
If we pivot from C in the opposite direction on our piano keyboard (or our scale carried out over several octaves), and we count in descending order seven steps to the note that forms the other perfect fifth that can occur in the key of C major (using a C as a note on one end of the interval), we land on the note F - seven semitones from C. So if we use our home note (C) as the anchor point and count a perfect fifth descending, we get the fifth interval formed with the notes F and C.
So if the interval F to C is a perfect fifth we know those notes will blend together in a harmonious manner. We know that using C as an anchor point in the key of C Major, we can reach a perfect fifth using the C in two and only two ways - ascending the scale by 7 semitones to get to G, or descending the scale by 7 semitones which lands us on the other option - F. So if the relationship between the C and F can also form a fifth interval, it makes sense that leaving C Major and going to a chord with a root based on F (F Major - our IV chord) or going from an F back to C, will sound natural in the key of C Major.
To further illustrate why the IV and V chord segue well with the tonic I chord which anchors the key, I might point out that the next most consonant interval between notes is the perfect fourth with a sonic frequency ratio of 4:3.
If you start with the triad that forms the tonic chord of a given Major key, and use the root note of that Major triad as the anchor point, there are two notes you can reach that will each form BOTH a perfect fourth interval AND a perfect fifth interval using your home note of the key (which is the root note of the tonic chord) as one end of the interval.
Again using the key of C major for illustration, the interval between C and G is a perfect fifth and the interval between G and C (same two notes now inverted) is a perfect fourth. Thus using C as an anchor point, and G at the other end of the interval, you can form both a perfect fifth (if you ascend from C) and a perfect fourth (if you descend from C). Remember, these are the two most consonant (pleasing sounding) intervals available outside of the octave and unison.
You can also achieve this same feat using one other note - F. The interval between F and C is a perfect fifth and the interval between C and F (ascending the scale) is a perfect fourth. Again the C is the home note or anchor point.
This provides further logic to explain why chords with root notes based on the two notes in the diatonic scale for a particular key, that are each capable of combining with the home key note to make both a perfect fourth and perfect fifth, will be the most stable sounding chords in that key and will naturally evolve from or resolve to the tonic (I) chord that anchors the key.
To understand why the next safest bet – harmonically or mathematically speaking is the vi (minor sixth) chord of the scale we must understand that every major key has a relative minor key that contains the same notes (only starting at a different place when written out or played as a scale).
The relative minor key of any major key – is based on the sixth degree vi (minor sixth) of the major scale. The relative minor key will contain exactly the same notes as the major key it is related to. So in any major key, the relative minor key (based on the sixth scale degree) will contain the same notes as the root key our song is in, and the minor sixth chord (vi) will consist of notes that are in the key of the song and will therefore harmonize well with corresponding notes in the melody.
The fact that the vi chord is derived from the relative minor scale of the root key of the song, perhaps further explains why it seems to work well for composing a chord progression for any popular song one wishes to write.
It may take some time to digest this but hopefully it helps. If any more knowledgeable members of the community care to comment on, or edit my answer for the sake of clarification or accuracy, please feel free to do so.