A statistical study of inversions (slash chords) in popular music.

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This article is Part 3 of a multipart series looking at the statistics gathered from 1300 choruses, verses, etc. of popular songs to discover the answer to some interesting questions about how popular music is structured. Click here to read Part 1.

In this article, we’ll continue our exploration into the patterns evident in the chords and melody of popular music. First we will look at the relative popularity of different inversions (e.g. a C/E chord vs. G/B, etc.) based on the frequency that they appear in chord progressions found in the Hooktheory Analysis Database. Then we will take a statistical look at how inversions are most often used. For example, if an inverted chord is found in a song, what can we say about the probability for what the next chord will be that comes after it? This will be compared with how the non-inverted counterpart of the chord is used (e.g. a C/E vs. a C).

Inverted Chords

When a song is using a C major chord, the lowest note (often played by the bass player if there is one) is usually a C. Sometimes this is not the case, however, and one of the other notes that make up the C major chord will be played instead (the E, or G). These so-called inverted chords occur frequently in popular music. Guitarists reading tablature will recognize them as “slash chords” (i.e. C/E) where the note below the slash is the new bass note.

The Popularity of inversions of C and G

Any chord can be inverted, but in practice it turns out only a few are commonly used in popular music. Amongst songs that had inversions in them, the following plot shows the rate at which different inverted chords show up in the database. All songs were transposed to the key of C to make comparison between songs in different keys valid.

The plot shows that C/E and G/B chords (I6 and V6 in Roman Numeral notation) are the most common by far. Part of this can be explained by the fact that, as we learned in Part 1, I and V chords are very popular chords in general. Yet we also learned that IV and vi chords are almost as popular, and inversions of these chords show up much less frequently than you might expect.

Perhaps a more interesting question to ask is, given that C/E and G/B (I6 and V6) are so popular, how are these chords used relative to their non-inverted vanilla C and G chords? In other words, how does changing the bass note of the chord change how the chord functions?

This question is readily answerable with empirical data from the Hooktheory database.

The usage of C/E vs. C

Let’s start by comparing C/E and C. The following plot below shows the probabilities for what the next chord in the chord progression will be after each chord. The first plot shows the probability for the chord coming after a vanilla C (I) chord. The second plot shows how the probability changes when this chord is instead a C/E (I6) chord.

The most common chords to follow a vanilla C (I) chord are unsurprisingly G (V) and F (IV) (occurring 26% and 46% of the time respectively). The tendencies change dramatically for C/E, however, with F (IV) being by far the most common chord and Dm (ii) gaining quite a bit of ground. Dm is almost 3 times more likely to come after a C/E chord as compared to a vanilla C chord (23% vs. 8%).

These statistics reveal an important function of inverted chords. Inversions are often used to link the bass notes between neighboring chords. In this case the E in the bass of C/E is a neighbor to the D and F in the bass of Dm and F respectively. This is very much in line with what a classically trained musician would expect, but it’s interesting to see these chords used in this way given that many popular songs are composed by people without classical backgrounds. The reason this is such a powerful technique is that our ears hear the lowest note in a song very strongly. It is perhaps less strong than the top note which is why melodies are so important, but the bass note is definitely critical (bass players can take solace in this fact).

A great example of a song that uses the I6 in this way is the intro of Michael Jackson’s Man in the Mirror. The song floats between ii (Dm) and IV (F) using I6 before finishing in textbook fashion with IV → V → I.

The usage of G/B vs. G

We find a similar story for G/B (V6). The following plot shows the probabilities for what the next chord in the chord progression will be after each chord. As before, the first plot shows the probability for the chord coming after a vanilla G (V) chord. The second plot shows how the probability changes when this chord is instead a G/B (V6) chord.

The plot shows that vanilla G (V) chords normally split almost evenly between going to C, F, and Am (I, IV, vi) (32%, 25%, and 29% respectively). Again since these are the most common chords, this should be not be all that surprising. However, the data clearly show that this is no longer the case for G/B (V6). Going to the vi doubles in popularity (from 29% to 61%) to become far and away the most popular transition. F (IV) takes a big hit, occurring 2.5 times less frequently, while C (I) holds more or less steady. Why does G/B to Am seem to occur so much more than G to Am? It’s likely for the same reason that C/E likes to go to F and D so much. Linking up neighboring bass notes creates a nice effect that clearly is a very common technique in popular music.

A good example of a song in the database that uses G/B like this is the Rolling Stone’s “Beast of Burden”:

Here are some other examples in the Song Database that do this too:
“Freebird” by Lynard Skynard
“You’re Beautiful” by James Blunt

“Your Song” by Elton John is a song that does something uncommon after G/B. It goes to the iii (Em). Aside from this departure, Elton tends to use chords in a very typical fashion (though he’s uses every trick in the book to get the sound he wants).

Though he doesn’t use the V6 to create a linked bass line here, he uses inversions of 7 chords (vi42) and secondary dominants (vii˚7/V, oh my!) to connect the vanilla vi chord by step with the IV to complete the progression (the bass note goes A → G → F# → F. We’ll talk about how songwriters like Elton use these more complex harmonies in future posts.

What’s your favorite use of inverted chords in a pop song? Are they used in a way consistent with these findings? Let us know in the comments below.

Other posts in the 1300 song series


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  • Ben

    Really good work. I love reading this stuff!

  • Ben

    great work guys! Love reading this stuff.

  • http://twitter.com/Stigjohan Stig Johan Berggren

    The Daft Punk tune ‘Superheroes’ has some excellent use of dominant 7th chords in the third inversion to create a chromatically descending bass line. It goes G – A/G – D/F# – G/F – Em7 etc.

  • Pingback: Part 2: I analyzed the chords of 1300 popular songs for patterns. This is what I found. | Blog – Hooktheory

  • Pingback: I analyzed the chords of 1300 popular songs for patterns. This is what I found. | Blog – Hooktheory

  • theReverbSignal

    Here’s a cool idea for future posts: while reading this, I was like, okay cool, C/E goes to F because of the bass note, gotcha. But then, to make sure I got it down pat, looked up C/E in order to compare it to F and got this:

    http://www.guitarchords247.com/chords/C-E

    So, just for clarity, chord charts telling us exactly which version of C/E you’re using (and I guess also which version of F) would be handy to visualize the change you’re talking about.

    But regardless, I love the way you’re taking a statistical analysis of music. It’s still amazes me how much people rely on I ii iii IV V and vi chords (and the slash variants). Although it would be super interesting to see the patterns in the way people use chords to break key for a moment. I’m guessing it’s more common in songs using a minor key, but exactly how is a question I haven’t yet found a good answer for. There are so many seriously interesting questions one could answer with this database!

    • http://www.hooktheory.com/ Dave

      Great thoughts here. Just to clarify what we did, our analysis ignored the so-called “voicing” of the chords so if the chord was a C major chord with an E on the bottom, we analyzed it as C/E regardless of how the other notes in the chord were arranged. Usually, the bass note is much more important for determining the function of the chord than the exact arrangement of the notes in the middle. That said, the different voicings definitely impart a slightly different sound so they aren’t unimportant.

      The use of chords “outside” the key is indeed a topic that we would like to cover in a different post. This is definitely a rich area with a lot to discuss.

      Thanks for the thoughtful points.

  • http://www.facebook.com/suyapc Suya Pereira Castilhos

    I love the sound of “C – C/E – F” in U2 “stuck in a moment you just can’t get out of”, the whole song is a good exemple of how the bass note can change a chord role in the song.

  • Russell

    Most popular modulations…?

    • Alex

      My guess would be flattening the fifth like in blues. So because G is the dominant, probably a G to an f sharp. Along with flattening the seventh.

  • Jamie

    I’ve been analysing songs and chords (and writing music) since I was about 6. Even at that age, I decided I would use my keyboard to transpose EVERY song into C major or A minor, so that my understanding of chord progressions would be twelve times more acute!

    Given my chord-obsessive background, I can certainly appreciate where this article is coming from, and the type of analysis you have tried to make here.

    However, in my opinion (and with great respect) this article doesn’t actually teach people anything about song writing. It’s rather like making a list of the most popular colours used in great paintings. It gives you no insight into the creative process or the subtleties of creating an artistic work. Just a bunch of superficial facts about the finished work. When I write music there are SO MANY different factors and artistic parameters that affect my choice of chord progression. My most fundamental rule is to NEVER do “the norm” and always try to innovate.

    While I know what chord sequences “work”, and am fully familiar with all the popular variations on sequences, for me, all this information can only ever be used as GUIDELINE in setting out on composing a creative work. In a sense, my knowledge of chords serves to warn me what NOT to do, as it has already been done to death.

    Another point I want to mention is chord types. Your analysis mostly only looks at major and minor chords, when in fact (as you are obviously aware) there are dozens and dozens of different chord types (counting inversions). So if you were to study ALL chord types from a purely statistical point of view, analysis would be a vast and futile affair. There are just too many different types of chords and different factors to study statistically. Whereas if you have a lifetime of experience, and a creative ear, then you can start to use chords in a way that has artistic merit.

    You may say that a study of ALL chord types is irrelevant since the majority of popular songs use simple chords. I would argue that if you are to rule out more complex chords from any equation, then you are imposing creative boundaries, and a creative work should be free and unlimited. When I write a song, I write the notes and chords my soul wants to hear. This may in rare cases be a very plain, predictable chord sequence; in other cases it may be a chord I’ve never used before but just sounds right; then again, it may be a deliberate discord to add tension to the piece; or it may even be a sound that defies the pentatonic scale and uses “weird new notes”.

    So, overall, thanks for writing the article. But I really don’t think statistical analysis can in any way assist music composition. I believe music lives in our souls, and a musician is someone with an ability to tap into that, extract it, and recreate it fully in all its glory. This process can be aided by experience, skill with an instrument, and technical knowledge; however, not by statistical analysis.

    • David

      You said: “While I know what chord sequences “work”, and am fully familiar with all the popular variations on sequences, for me, all this information can only ever be used as GUIDELINE in setting out on composing a creative work.”

      Then perhaps you should see this article as something of a GUIDELINE for others who haven’t spent as long as you analysing chords. A shortcut if you like to seeing empirically what works and what doesn’t.

    • http://www.facebook.com/RobinFrench Robin French

      Hey there, don’t suppose you have time to listen to my music. Any insight or feedback would be great. http://www.southsecond.bandcamp.com

    • Pete

      Joe Pass used incredible display of Chord variations yet he saw every chord as either Major, Minor or Dominant.
      The best musicians reduce complex issues to simple thoughts with, as you said, deep soul.

  • http://www.facebook.com/juan.disko Juan Disko

    Really good work man! :)

  • Jose

    I certainly enjoyed the read of the series and is a great exposure of facts. The whys and because I think are a bit trickier. Something not mentioned in any of the articles is how the geometry of an instrument affects the chord choices. I’m sure that guitar songs and piano songs analysed on their own would have slight (or not!) differences, just because how easy it is to play certain chord on certain instrument, or going from one chord to another. This would be reflected not only in chord choices but also in inversion choices and extended chords as well (there’s so many guitar players that don’t even know the chord their playing when they add or remove a finger from a string!).

    I would love to read a continuation of these articles having this into account.

    Great work, thanks a lot!

  • Phil

    In the newest movie Iron Man 3 they used the C major chord for Iron mans theme song. I understand the C scale is the most popular mainly because it is it is the most easy to base a song off. When writing music I feel that C scale is an easy place to start. However this is just my opinion and my feelings.