13

Over time, many musicians have experimented with various chord progressions, but none has stood the test of time as the four-chord song. What is it about this progression (and its variations) that are so appealing to the masses? Is there any mathematical basis behind it?


  1. That I asked for mathematical basis does not mean this is the only answer. Subjective answers (with reasoning) are encouraged as well.
  2. So as to keep answers short(er), assume the reader (myself included) is familiar with chord theory and terminology (i.e. I=tonic, V=dominant, etc). No need to reinvent the wheel.
  • Oh - so what you are asking is "why are 4 chord songs containing the I, V,iv and IV chords so prevalent in popular music?" Is that correct? – Rockin Cowboy Feb 13 '15 at 1:33
  • 2
    When I think of riff, I think of a lead guitar lick or solo - not chord progression – Rockin Cowboy Feb 13 '15 at 1:35
  • I'm working on an answer based on your desire to understand the mathematical basis for this? Hopefully I am on the right track. It's not a short answer. Are you okay with that? – Rockin Cowboy Feb 13 '15 at 1:52
  • I'm composing my mathematical answer now. Stand by. – Rockin Cowboy Feb 13 '15 at 3:14
  • You linked to a wiki article about the I, V vi IV progression so my mathematical explanation is specific to use of those chords but will probably give you all the math you care to try to absorb in an evening. – Rockin Cowboy Feb 13 '15 at 3:33
13

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.

6

The first thing that I would point out is that within a given key, you can harmonize any diatonic melody using only three chords, I, IV and V.
Key of C
I = C = CEG
IV = F = FAC
V = G = GBD
You can see that all 7 notes of the scale appear in at least one of these chords, allowing you to consonantly harmonize each scale degree. The classic diss on Rock music comes to mind, "You only need to play three chords and scream".

This can be slightly different in a minor key. If you are playing a functional minor, ie, your V is a major/dominant chord, as opposed to playing modally, then the 7th degree of your scale is altered to become a leading tone when V is played. This means that the natural 7 of minor (b7) is not covered by the V chord and would need to be harmonized otherwise, such as with the (b)III or (b)VII chord. If you are playing modally (not altering the 7th degree of the scale on your V chord), then the concept remains true that all scale degrees can be harmonized by the i, iv and v chords. I like to point out that in both cases the chords that will consonantly harmonize all diatonic notes are built on scale degress 1, 4 and 5.

I like to think of the 3rd chord that appears in the 4 chord progression as a texture change. The 3rd chord is your vi chord, which is the relative minor. I and vi share two notes (scale degrees 1 and 3) and vi is known to be a substitute for I because of this relationship. Additionally, vi and IV share two notes in common (scale degrees 6 and 1), so vi can also be used in places where IV could be used, though slightly less common as a substitution.

So I think of the I-V-vi-IV progression as I-V-substitute for I (vi)-IV. Pretty much any of the tunes that are played with this progression could be reharmonized to have I in the place that vi appears, I-V-I-IV.

I'd also point out the plagal cadence (IV-I) the progression creates. I feel that this makes such a song's repetition flow a little smoother because the resolution of IV to I is less strong than V to I. The resolution being less strong makes each phrase feel less complete, almost asking for more music to follow. I also sometimes think of the progression as feeling somewhat modal, which would be the Lydian mode, where the IV chord feels rather resolved to me, though I'm pretty sure most would classify this as being in the major key or, if modal, the Ionian mode.

As for the math side of things, I believe that chord theory and harmonization theory cover that. I don't find anything about it mathematically relevant beyond the basic concepts used to construct our harmony based idiom.

  • i also see a "texture change" - interesting idea there. @rockincowboy called it building tension. shared notes is definitely a factor i think, but the counterargument is why the two shared notes in iii each with I and V or the two shared notes in ii with IV don't make those chords appear more frequently in pop chord progressions. or do they? (i see I-IV-ii-V and variations thereof as the second-most common 4-chord progression.) need some historical analysis of popular music i think. – Erich Feb 26 '15 at 22:50
  • That might be a whole other question. I believe that iii is used less frequently because it is more dissonant in nature. The 1/2 step up to the second degree is part of this and the leading tone being a part of the chord is another. This is more of a dissonance within the key, than of the chord itself. As you mention ii is another common chord in the mix, it is also used as a predominant, so the use of ii following IV could be analyzed as an extension of the predominant. – Basstickler Feb 26 '15 at 23:10
  • Typically a chord that would be used to substitute for another when the second chord has the root in common, eg, Dmin substitutes for F because Dmin shares two notes, one being the root of the original chord, F. In my answer I intentionally did not say that vi is a substitute for IV, more that the notes they share allow for a choice to be made, which could end up coming down to the function you are looking for. vi does not have the same function as IV, so it is not exactly a substitute. – Basstickler Feb 26 '15 at 23:14
5

Now for the short answer.

As it occurred to me that not everyone will want to read my dissertation in the first answer, I will attempt a simpler explanation for why many popular songs are based on a similar four chord progression.

This answer will still explain it mathematically as suggested by the question.

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).

Music in common usage in most of the Western World today divides an octave into 12 steps. Each key contains 7 of the 12 possible notes. The distance between these notes is known as an interval and is measured in steps (also known as semitones). Each octave has 12 semitones but the intervals between them define which notes are in that scale and if the scale is major or minor.

The harmonic quality of an interval between notes (or how pleasing it sounds) is determined by the ratio of the frequency of each notes sound waves to one another.

So two people singing the exact same note at the same time will produce a sound that blends together well. We call this interval a unison and it has a ratio of one crest of sound wave to one crest of sound wave or 1:1.

The next closest aspect ratio of crest to crest that we can achieve is the octave which is the note at exactly double the frequency of the first - or 2:1. An example of an octave interval is the bass strings on a 12 string guitar are tuned an octave apart, so one vibrates exactly twice as fast as the other so the crest of the waves coincides one for every two making those two notes at that octave interval blend well together and sound good.

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 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.

The next most consonant interval between notes is the perfect fourth with a sonic frequency ratio of 4:3.

In choosing chords for a song in a particular key, the chords most likely to support the melody will consist of those chords that contain the notes of the scale corresponding to the key the song is in. Those notes give us only three major chords if we are composing an upbeat song in a major key. And those 3 options for major chords will always end up being the I chord, IV chord and V chord (all major) because of how a chord is formed.

The I chord is formed using the tonic note of the key (C in the key of C) as the root note of the chord. So it's a major chord based on the first degree of the scale corresponding to our 1:1 unison. The IV chord is the major chord based on the fourth degree of the scale which relates to our aforementioned fourth interval at a 4:3 ratio and the V chord is based on the fifth degree and thus is related to our 5th interval at a harmonically pleasing ratio of 3:2.

Some other chords available to us for our chord progression that also consist of notes found in our key, are the ii (two minor) based on the second degree of our scale, the iii (three minor) and the vi (sixth minor).

Since the sonic frequencies of these minor chords in our major key will be less congruent than in our 3 major chords (the I, IV and V) they will create tension in our arrangement. This tension will create a desire for resolution back to a major chord and ultimately back to the tonic or I chord.

They are available to us in our chord progression since they consist only of notes in our key. And if the composer wants to create an element of tension in a part of the song, one of these minor chords will often be added to the I, IV and V chords to create a four chord song. A fourth chord will sometimes make the song more interesting with the tension and resolution creating a dynamic sense of movement in the music. Kind of like a push pull effect.

But the chords available to increase the probability of widespread appeal in a song, will be limited primarily to the chords discussed above. These are the chords who's notes and intervals as they relate to the sonic frequency ratios between them - will blend most harmoniously and pleasingly with the notes in the melody of any song in a given key.

For a more detailed discussion of this, see my other answer to this question.

  • Okay so the short answer is still not exactly short. But it's a difficult concept to explain mathematically. And a proper academic explanation of harmonic overtones and such could easily become an entire textbook. This was my attempt to show a simplified mathematical explanation for why certain chord progression are so popular. I welcome edits, comments, corrections, etc. – Rockin Cowboy Feb 13 '15 at 6:06
  • i like the idea of adding tension. however, if the only qualifier for these minor "tension-building" chords is having the same notes as the tonic major scale, why is vi way more common than ii or iii? – Erich Feb 13 '15 at 6:15
  • 1
    Also the ii and iii minor triads do contain the notes of the key the song is in, however the full 7 note scale for the key corresponding to the 2nd and 3rd scale degree of the key you are in, will contain notes that are NOT in the key your are in so any lead solo's that were based on the chord you were playing at the time could not use all the notes of the scale of the ii or iii chord, nor could the melody use all of the notes of those scales. But if you sing or play fills or riffs over the vi chord you can use any of the notes in the tonic key the song is in and they will go with that chord. – Rockin Cowboy Feb 13 '15 at 8:37
  • 1
    I think this is a good summary of the basics, and any chords not obtained by these methods typically share at least one or two tones with a 'mathematically desirable' chord (and they very frequently contain leading tones - half step intervals away from - other 'in-key' chords). – Darren Ringer Feb 14 '15 at 0:02
  • 1
    @mey Thanks. I think I might kill the short answer and make the long answer longer by adding anything in Short that is missing in long. Erich the OP is not happy that there are no other answers. He edited to appeal for more answers and opened it up to subjective answers as well. Feel free to take a stab at it from an objective POV if you like. – Rockin Cowboy Feb 17 '15 at 23:12
5

(Pardon me if I explain things you already know, but I'm aiming a little lower than the questioner asked for in terms of compositional understanding, in hopes of helping other people who stumble across this page and don't know much about how harmony works.)

The progression works because of voice-leading. The voices in this progression barely move, and they move in small and simple ways.

Voice-leading refers to the composition of individual melodic lines within a larger, harmonic composition, so that the melodies move to pitches that sound pleasing one after the other. In other words, in a choral setting, where each person sings one pitch at a time, the ensemble sings together in harmony, making chords out of the simultaneous pitches that they're singing. The individual melodies that each singer sings still sound good by themselves, without a lot of surprising leaps from pitch to pitch.

J. S. Bach was extremely good at this, and if you listen to his choral works (and look at the sheet music, even if you don't read music notation), and focus on one singer at a time, you can hear what's happening. Every voice is doing something nice and not too crazy, nor are they clashing against each other, so everything just sounds good all the time. If you want to learn more about how composers achieve this, look into "counterpoint", which you can learn using a technique called "species counterpoint".

On a polyphonic instrument like a guitar or piano, you can sound multiple simultaneous pitches to form chords, but it still sounds best if you consider how a series of chords will work from a voice-leading point of view. A thing that you can do to chords to improve the voice-leading awesomeness of your compositions is to use inverted chords.

So, if you write a song with a I-V-vi-IV progression, you can make the voice leading really, really simple, so that each voice is moving just a teeny bit from chord to chord.

In the key of C, it would be C-G-Amin-F, but if you use the second inversion of C and the first inversion of F, (C/G - G - Amin - F/A) you get something much nicer. I'll write out what the chord tones are and explain why it sounds good, and I'll include the resolution from the IV back to I at the end so it's even clearer:

Without inversions:

C:    C E G   I root position
G:    G B D   V root position
Amin: A C E   vi root position
F:    F A C   IV root position
C:    C E G   back to I

With inversions:

C/G:  G C E   I, 2nd inversion
G:    G B D   V root position
Amin: A C E   vi root position
F/A:  A C F   IV, 1st inversion
C/G:  G C E   back to I, 2nd inversion

The lowest voice goes G-G-A-A-G, the middle goes C-B-C-C-C, and the highest voice goes E-D-E-F-E.

Written as scale degrees in the key of C, that's 5-5-6-6-5, 1-7-1-1-1 (think of it as 8-7-8-8-8), and 3-2-3-4-3.

So, the bottom voice only moves by one scale degree and then back, the middle voice only moves by one scale degree and then back, and the top voice goes down one, back, up one, and back.

From a harmonic standpoint, the I-V-vi motion is called a "deceptive cadence" because G-C-E and A-C-E are so similar: you almost did a I-V-I but faked out the listener and landed on vi instead. Then, you moved one voice by a single step (E to F, changing the Amin into an F/A) and you're on IV. These small changes make chords flow into each other smoothly from the listener's point of view.

Another inversion that works is I, the first inversion of V, the first inversion of vi, and the second inversion of IV. In C that's C - G/B - Amin/C - F/C.

C:      C E G   I root position
G/B:    B D G   V, 1st inversion
Amin/C: C E A   vi, 1st inversion
F/C:    C F A   IV, 2nd inversion
C:      C E G   back to I root position

So the voices go C-B-C-C-C, E-D-E-F-E, G-G-A-A-G. In scale degrees of C that's 1-7-1-1-1 (think 8-7-8-8-8), 3-2-3-4-3, and 5-5-6-6-5 again.

This one is the one that's shown in the Wikipedia article for the I-V-vi-IV progression. https://en.wikipedia.org/wiki/I%E2%80%93V%E2%80%93vi%E2%80%93IV_progression

This is probably a little nicer than the one above, because the I chord is not inverted, meaning that the lowest note the listener hears is almost always C, which will really give a strong sense of everything being in C.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.