What are all the possible ways to voice a seventh chord?
This post is a quick note on that question, with a focus on how a guitarist might think of things. I’m posting this so I can refer to it later.
To start out, we have the root position voicing. I’m going to denote this as 1357: the root at the bottom, then the third, then the fifth, and finally the seventh on top.
Next, we can take all the inversions of the root position voicing. First we move the root up an octave, leaving the third in the bass. This gives 3571, first inversion. Next we move the third up an octave, leaving the fifth in the bass. Now we have 5713, second inversion. Finally we move the fifth up an octave, giving 7135, third inversion.
So far, we’ve considered four voicings. It turns out that many of these “close” voicings are difficult or impossible to play on guitar, but things get much easier to finger if we take these four voicings, identify the second note from the top in each of them, and drop it down an octave. If we begin with the root position voicing, 1357, then the second note from the top is the fifth of the chord; dropping it gives 5137. If we begin with the first inversion, 3571, we’ll be dropping the seventh, giving 7351. Similarly, we get 1573 and 3715 from the second and third inversions, respectively. These four voicings are called the “Drop 2” voicings, with the “2” meaning “second from the top.”
Sometimes you might hear the Drop 2 voicings referred to as Drop 2 “inversions,” but don’t be confused by this. The Drop 2 voicings are not inversions of each other. If we were to take the first Drop 2 voicing, 5137, and invert it by moving the bass up an octave, we’d get 1375, but notice that 1375 isn’t anything we’ve listed so far. The Drop 2 voicings arise when you take the root position chord and its three inversions, and then you apply the Drop 2 process (dropping the second highest note) to each of those close voicings. They are not created by starting with the first Drop 2 voicing, 5137, and taking its own inversions, which would give different results. Subtle point but worth noting.
Just like we created four Drop 2 voicings, we can create four “Drop 3” voicings by taking the root position chord and its inversions and dropping the third note from the top. You’ll find that we can get four more voicings by Dropping 2 and 4, and another four voicings by Dropping 2 and 3.
Any other scheme you might come up with for dropping notes is going to yield voicings we’ve already seen. I’m not going to prove that formally here, but you can try some possibilities. Let’s say we take the root position chord, 1357, and drop the highest note to create a “Drop 1” voicing: it turns out to be 7135 which is just the third inversion of the root position chord, old news. Of course, if we try to create a “Drop 4” voicing by dropping the lowest note, we get the same voicing back: 1357 is still 1357 with the root moved down an octave.
In the book The Advancing Guitarist by Mick Goodrick, on Page 44 of my copy, he mentions five sets of voicings: the close voicings, the Drop 2 voicings, the Drop 3 voicings, the Drop 2&4 voicings, and the Drop 2&3 voicings: this is everything we’ve just considered. These 5 sets of voicings each have four members, giving a total of 20 voicings. Have we covered all the possible voicings or are there others we haven’t considered?
Simple combinatorics tells us that if we have 4 items (in our case, four notes) there are 4-factorial ways of permuting them. 4-factorial is 4*3*2*1 which is 24, but we’ve only identified 20 voicings so far. It seems there must be 4 voicings we haven’t considered yet!
It turns out that there are several voicings that you just can’t derive by starting with 1357 and its inversions and dropping notes down an octave. I call these four “missing” voicings the Reverse Voicings. Take the root position chord, 1357, and reverse the order of the notes: you get 7531. Now take the first inversion of the root position chord, 3571, and reverse it: you get 1753. Likewise, you get 3175 and 5317 from the second and third inversions, respectively. None of these arise from any kind of “dropping” process, but they’re still legitimate voicings. Having identified these Reverse Voicings, we’ve now got 24 voicings and combinatorics tells us that no more are possible.
When you play around with voicings, some interesting relationships between the different sets become apparent. For example, we can turn Drop 2 voicings into other voicings by moving notes up and down:
If you take a Drop 2 voicing and move the lowest note to the top, you get a Drop 2&3 voicing. For example, 5137 becomes 1375.
If you take a Drop 2 voicing and move the highest note to the bottom, you get a Drop 3 voicing. For example, 5137 becomes 7513.
Move the second highest note to the top → Drop 2&4 voicing.
Move the second highest note to the bottom → Drop 2&3 voicing.
Move the third highest note to the top → Drop 3 voicing.
Move the third highest note to the bottom → Drop 2&4 voicing.
Here’s a table of all the 24 possible voicings:
|Close||Drop 2||Drop 3||Drop 2&4||Drop 2&3||Reversed|
|Root or root derivative||1357||5137||3157||1537||3517||7531|
|1st inversion or derivative||3571||7351||5371||3751||5731||1753|
|2nd inversion or derivative||5713||1573||7513||5173||7153||3175|
|3rd inversion or derivative||7135||3715||1735||7315||1375||5317|
I’ve often thought that a good, comprehensive way to compare the potential of different guitar tunings would be to take all the types of seventh chords, and for each type, consider its 24 possible voicings. For each voicing, look at how it falls on the fretboard: is it playable in the given tuning? This would give a kind of encyclopedia of what’s playable in each tuning. I’ve been contemplating this exercise as a way of comparing P4 and M3 tuning, but it’s been more work than I’ve been ready to commit to. Interested in alternate guitar tunings? Want to help?