# Why do we have inversions but not permutations?

So if I have the C major chord, the number of permutations it has is 3! (3 factorial) = 6.

1. CEG
2. CGE
3. ECG
4. EGC
5. GCE
6. GEC

But when they talk about inversions, I only have three choices:

1. CEG
2. EGC
3. GCE

So are permutations used? If so, is there a musical term for them? If they're not used then why are inversions "more important" ?

• This is sort of mentioned in ttw's answer and implicit in Todd Wilcox's answer, but I think not stressed enough: you do not "only have three choices" for inversions in the sense that there's no name for the permutations that are left out of your list. All six permutations you've listed are valid inversions, it's just that half of them are considered the "same" inversions as the other half. E.g. "CGE" is considered the same inversion as "CEG" (assuming the first letter indicates the base-voice note). – Kyle Strand Apr 18 '17 at 19:12
• – Richard Apr 23 '17 at 19:03

Psychoacoustically, changing the lowest sounding note of a triad makes a bigger difference to how we hear that triad than changing the order and number of the other notes, so we talk about inversions (voicings where the lowest note is not the root) a lot more.

As Dom points out, there are more than six voicings for a triad, because notes can be duplicated. There are a large, but finite, number of voicings for any triad. Here are some examples:

Root position:

• CEGC
• CGECE
• CCEGCE

First inversion:

• EGCE
• EECEG
• ECGCEG

Second inversion:

• GCEG
• GEGCE
• GGEGCE

It's pretty popular on piano to play octaves on the left hand, doubling the lowest sounding note, and then play three- and four-note voicings for triads on the right hand.

Voicings are pretty important on guitar since you are limited to six notes maximum and you have to be able to reach all of the notes in one hand span and fret them all with four fingers (or play them open). As an Algebraist (who is fascinated with permutations) and a guitarist (who must think about voicings), I find the concepts to be basically identical.

• Those missing inversions that OP is calling "permutations" are often called drop voicings. Drop 2 voicings are very common for guitarists. – wim Apr 18 '17 at 2:54
• Bang on. If only every question on StackExchange could get an answer as well-reasoned and clear as this. – Matt Taylor Apr 18 '17 at 10:12
• I don't think I'd still call EECEG a voicing of C-major anymore, though. – leftaroundabout Apr 18 '17 at 15:47
• @leftaroundabout Good point. It's an edge case to be sure. On the other hand if the CEG is in the right octave with the right loudness it might come across as c major in inversion. – Todd Wilcox Apr 18 '17 at 16:54
• @leftaroundabout: I'd call that a voicing of C major. More interesting, though, would be whether something like E E G Bb E could interpreted as a C7 when followed by F F A C F, since the most important parts of a dominant 7th are the third which resolves up a half step and the 7th which resolves down a half step, and four of the notes in E E G Bb E serve those roles. While the chord might more accurately be regarded as a diminished vii than V₇, the root of a V₇ isn't what creates its essential character. – supercat Apr 18 '17 at 18:28

It's a convention going back to medieval times. Chord inversion is based on the bass note. Thus CEG and CGE orderings are named the same as are EGC and ECG and the pair GCE and GEC. When thinking of a bass line, this makes some sense. It works in more modern music especially homophonic styles. There is a bass line on the bottom, a melody on top, and a bunch of chord-like stuff in between. I would say that the harmonic difference between CEG and ECG is much greater than that between CEG and GCE.

• The concept of "inversions" only goes back to Rameau's textbook on music theory published in 1744, not to "medieval times". Before Rameau, "inversions of a chord" were considered to be different chords. Chords were classified by counting intervals from the root, and in that sense all seven "root position" chords in a major scale were considered "the same," regardless of the fact that they were major, minor, or diminished. See CPE Bach (writing just after Rameau had published the concept of "inversions") for example. – user19146 Apr 17 '17 at 21:29

In general, you can think of a chord as a set of notes. The exact number of notes and the exact ordering of the notes does not matter and the only thing that matters for the inversion is the bass note. So in your example, 1 and 2 are both root position, 3 and 4 are both 1st inversion, and 5 and 6 are both 2nd inversion.

The permutations can be viewed as a specific voicings, but honestly it's very rare you'll actually look at voicings this way outside of set theory due to it missing a big aspect of voicing which is notes can be double and typically are hence the permutations alone can't represent this very well. To talk about a vocing in any meaning, you'll need to know exactly where each note falls in the chord and representing in another form such as staff makes it much easier to talk about.

All of these answers are great, but I'm going to go a different direction and address how permutations actually can be used!

There's a branch of music theory called neo-Riemannian theory, and it looks at what we call "parsimonious voice leading." ("Parsimonious" basically means "most efficient.")

Let's say we have a C major triad that moves to an A minor triad. Here's one way:

``````G  E  (down three semitones)
E  C  (down four semitones)
C  A  (down three semitones)
``````

Between these two root-position triads, then, we have a net movement of 10 semitones (!). Hardly parsimonious. So let's move to an A minor triad in first inversion instead!

``````G  E  (down three semitones)
E  A  (down seven semitones, or up five)
C  C  (no movement!)
``````

This is actually equivalent in a sense; if the E moves down to A, we again have a net movement of 10 semitones. Even if the E moves up, we're looking at 8 total semitones traversed.

So here's where the permutations come into play, because if we clarify the first-inversion A minor triad as `C E A`, we get:

``````G  A  (up two semitones)
E  E  (no movement!)
C  C  (no movement!)
``````

Here, only one voice is moving, and it by just two semitones!

So, TL;DR: Permutations can be used in music theory to discuss efficient voice leading, but this is a very different scenario than you were addressing in your original question. If any math fans are really interested in it, here's an article you can check out.

Inversions have always been classified using the lowest note as a datum point. Thus in triads, there can only be three - root/1st/2nd. It's how theorists did it. True, there are other voicings, which complicate the inversion aspect, and on some instruments - guitar in particular, uncomplicated voicings are sometimes not possible.So a new(ish) term comes into play- literally!

Check out DROP VOICINGS which are what you search for. They can get a little complex, as alluded by all your permutations, but everything you crave is there - in drop voicings!