Chord Inversion Master List

[JakeAcci]

I've come up with a very fast way to find all possible inversions of any four note structure. Bear with me...

Here are all possible ways to voice a seventh chord, with the root on the bottom, no doubling/octaves, and no interval larger than a seventh in between voices:

1357 (closed)
1573 (Drop two)
1735 (drop three)
1375 (drop two and three)
1537 (drop two and four)
1753 (double drop two and drop three)

Those forms plus their inversions:

close:


1357
3571
5713
7135


drop two:


1573
3715
5137
7351

drop three:


1735
3157
5371
7513

drop two and drop three:


1375
3517
5731
7153

drop two and drop four:


1537
3751
5173
7315

double drop two and drop three:


1753
3175
5317
7531

You can use the lists above to find the inversions of any four note structure. Copy the list into any word processing program, and use the "find and replace" feature. To find inversions of other seventh chords it's pretty easy to do it in your head...let's say Cm7b5 - 1=C, 3=Eb, 5=Gb, 7=Bb, but the find and replace tool makes it automatic:

close:



C Eb Gb Bb
Eb Gb Bb C
Gb Bb C Eb
Bb C Eb Gb




drop two:




C Gb Bb Eb
Eb Bb C Gb
Gb C Eb Bb
Bb Eb Gb C


drop three:




C Bb Eb Gb
Eb C Gb Bb
Gb Eb Bb C
Bb Gb C Eb


drop two and drop three:




C Eb Bb Gb
Eb Gb C Bb
Gb Bb Eb C
Bb C Gb Eb


drop two and drop four:




C Gb Eb Bb
Eb Bb Gb C
Gb C Bb Eb
Bb Eb C Gb


double drop two and drop three:




C Bb Gb Eb
Eb C Bb Gb
Gb Eb C Bb
Bb Gb Eb C

It works just as well for four note structures that are not tertial seventh chords, just replace 1, 3, 5, and 7 with whatever you are working with. For example if I wanted to find the inversions of a four note cluster, C D Eb F. I'll simply replace 1 with C, 3 with D, 5 with Eb, and 7 with F, and I get the below:

close:


C D Eb F
D Eb F C
Eb F C D
F C D Eb


drop two:


C Eb F D
D F C Eb
Eb C D F
F D Eb C


drop three:


C F D Eb
D C Eb F
Eb D F C
F Eb C D


drop two and drop three:


C D F Eb
D Eb C F
Eb F D C
F C Eb D


drop two and drop four:


C Eb D F
D F Eb C
Eb C F D
F D C Eb


double drop two and drop three:


C F Eb D
D C F Eb
Eb D C F
F Eb D C

Or I could keep it key-neutral and could have replaced as such:
1=1
3=2
5=b3
7=4

close:




1 2 b3 4
2 b3 4 1
b3 4 1 2
4 1 2 b3




drop two:




1 b3 4 2
2 4 1 b3
b3 1 2 4
4 2 b3 1


drop three:




1 4 2 b3
2 1 b3 4
b3 2 4 1
4 b3 1 2


drop two and drop three:




1 2 4 b3
2 b3 1 4
b3 4 2 1
4 1 b3 2


drop two and drop four:




1 b3 2 4
2 4 b3 1
b3 1 4 2
4 2 1 b3


double drop two and drop three:




1 4 b3 2
2 1 4 b3
b3 2 1 4
4 b3 2 1

[JakeAcci]

Three note structures have fewer inversions:

close
135
351
513


spread
153
315
531

But similarly, you can find the inversions of a three note structure that is not a triad by using the find and replace tool.

[JakeAcci]

Thanks to Bako, Mick Goodrick, and Jon Damian it's now clear to me that all four note structures within a seven note scale fit into one of these five classifications:

1. 7th chord (1 3 5 7)

2. Quartal stack (1 4 7 3 when arranged in fourths, 1 3 4 7 in close)


3. Four scale tones in a row - cluster (1 2 3 4)

4. A triad plus its fourth (1 3 4 5). Goodrick refers to this as a triad over bass note - I don't like that terminology because once inverted, that classification makes no sense. From my perspective it makes more sense to classify it as a triad plus its fourth, and can have four possible bass notes just as any other four note structure.

5. A triad plus its second.

What's fascinating to me is that if you look at any whacky four note structure (anything in that cluster thread I started, for example) you see that they all fit into one of the five above.

-----

Similarly, any three note structure within a seven note scale is either

1. A triad (135)

2. Cluster (123)

3. Seventh chord without a 3 (157)

4. Seventh chord without 5 (137)

5. Quartal stack (147) which can also be thought of as sus2 (125) or sus4 (145)*

*For example, DGC (quartal stack) is 147 from D, 125 (sus2) from C, or 145 (sus4) from G.

----

And though it will have less application to guitar, and I haven't personally gone too deep into it, any five note structure within a seven note scale is either

cluster (five scale tones in a row)…12345, which could be thought of as a triad plus its 9th and 11th


A 9th chord (13579) which in inversion is also 1 3 5 7 13


An 11th chord, as 1 3 5 7 11

---

Again what's fascinating to me is how few classifications there actually are - the inversion process changes the sound and texture of a structure so much! For example E C# A# G# is a m7b5 in inversion. E D# C# A# is a quartal stack in inversion.

[BigDaddyLoveHandles]

I didn't know there would be math


.
.
.
.
Seriously, I'll have to look at that when I'm fully awake.

[JakeAcci]

I didn't know there would be math


.
.
.
.
Seriously, I'll have to look at that when I'm fully awake.

Aww! There's no math! Just copy and past. I'm sorry, I didn't mean for the layout to look crazy. If numbers aren't your thing, below is the same list as a Cmaj7 chord, using note names rather than numbers. Then for a different chord, just use the find and replace feature and plug in the notes you want instead of CEGB.

close:



C E G B
E G B C
G B C E
B C E G




drop two:




C G B E
E B C G
G C E B
B E G C


drop three:




C B E G
E C G B
G E B C
B G C E


drop two and drop three:




C E B G
E G C B
G B E C
B C G E


drop two and drop four:




C G E B
E B G C
G C B E
B E C G


double drop two and drop three:




C B G E
E C B G
G E C B
B G E C

[NSJ]

Nice job. Duly copied and pasted. I confess I am well versed with close, drop 2 and drop 3. I think I've got them somewhat down. I suspect most of us do, and use these. I was just practicing some 3 note the spread voicings this morning, actually. I really like the sound of 1-5-3 (10ths). That and the 1-7-3 (10th) are bread and butter essential

But the other 3 (drop two and three, drop two and four, double drop two and drop three), yeah, not so much. and the last one: not at all.

Which begs the questions: what is the proper context to use the last 3 forms? And, generally speaking, when do you really want or need the 7 in the bass?

[jsepguitar]

Thanks Jake. I love this kind of thing.

As for the 3 and 4 note permutations, I do the same thing, but I use Excel (spreadsheet). I put the 3 or 4 notes at the top, and then have links below for the permutations so that when I change the 3 or 4 notes at the top it changes the permutations below.

[whatswisdom]

This is a keeper. Thanks for your hard work, Jake.

[JakeAcci]

NSJ, great questions.

But the other 3 (drop two and three, drop two and four, double drop two and drop three)... what is the proper context to use the last 3 forms?

I think there are a lot of ways to answer that!

I'm not sure of proper context to use, but proper context to explore might could be when you simply are bored and unsatisfied with the voicings you are already familiar with.

If your goal is to get through jazz gigs playing standards and not have it be a complete train wreck, then all that's truly needed is a handful of voicings, only a tiny fraction of what is actually possible. A few forms for each chord type, be able to put different notes on the top, and you're set, you get through the gig without everything falling apart.

If you are trying to make chord melody arrangements and are content to just play root position drop 2 or drop 3 chords and stick a melody note on top, then that's all you need. No need to bother with big spread voicings.

If you're not content to play that way, or to play in that context, then you look for other options, maybe things you find more interesting. I think many people find the process of discovering new vertical structures to be exciting and inspiring, and lead to new ways of playing, comping, composing, arranging, even playing lines.

Ben Monder and Mick Goodrick are two guitarists I can think of who can write and improvise on guitar in a way that is very sensitive to each individual voice when they are playing more than one note at a time. Ben Monder writes music for guitar and he plays as a sideman for many modern artists, and he makes use of many different voicings beyond drop 2, drop 3, and close. Another answer - he has to make use of these other voicings because he is often thinking of chord movement as individual line movement, and it doesn't always work out that the individual lines come together to result in familiar chord grabs.

More practically, the larger voicings cover a greater range of pitches, and I guess you could say may be better suited for solo playing, or playing without a bass player, but everything is context.

Personally, I want to make solo improvisations that sound like a string quartet, so, well, I have some work to do...

I've been looking at Debussy and Ravel string quartet scores and there is MUCH use of these larger spread voicings. Beautiful stuff.

If you take one chord and go through all 24 inversions you'll find that each one has a very different character and feeling...personally, some speak to me more than others, some of them seem more part of 'my story' than others.

Another answer:

Each inversion provides a way to harmonize a melody note. In the context of a solo arrangement, you might simply find one of the other voicings to be a better sound than one of the ones you are more familiar with.

Another answer:

Arpeggiating these, or any inversion, can lead to more material for linear playing.

I could go on. I think one of the exciting things about this stuff is, like Goodrick says in the intro to the first Almanac book, is that different people will make different discoveries with the same raw material. I'm exploring this stuff personally to help with my solo guitar improvisations and to become more aware of multiple voice improvisation, but I think somebody else may get other benefits based on what he or she is working on or trying to do, or what sounds he or she is after. For example, Tim Miller has a huge chord vocabulary, he improvises over standard tunes and writes his own material for trio and solo guitar, and well, listen to his music...

And, generally speaking, when do you really want or need the 7 in the bass?

I think my answer to this doubles a lot of the answers above.

The 7 in the bass in a higher register doesn't really act as a bass note at all...especially if you are playing with a bass player!

In a lower register it's still just another way to invert the chord or harmonize a melody (or four melodies simultaneously.) Consider voice leading, rather than the importance of a root on the bass...for example, G7 with F in the bass leads well to C with E in the bass.

Here's a ii V I voice leading example that uses 7th in the bass for the first chord, and some larger spreads:

Dm7: C X A D F X

G7#5b9: B X Ab Eb G X

Cma7: X C G D X B

the G7#5b9 is a drop 3 Abminmaj7 in second inversion

the Cma7 is a drop 2 and drop 4 G with a C note (triad with 4.)

Another significant thing is that a seventh in the bass of a 1357 chord means you actually have a triad on top of that bass note. Relative to the bass note, the triad is built form the second degree. Relative to the triad, the bass is the seventh degree.

So, you are dealing with triad over bass note harmony.

C/B is Cmaj7 in third inversion. But is also functions well as B7sus4b9 or as part of a B phrygian vamp, among other things.

C/Bb is C7 in third inversion. But it also can potentially be a voicing for Bbma7#11 or Bb7#11, or Bb lydian vamping (or lydian augmented or lydian dominant)

C+/B is Cma7#5 in third inversion, but that also works for B13sus4b9, second mode of A melodic minor

C+/Bb is C7#5 in third inversion, but a very common Bb9#11 voicing

Cm/B is Cminmaj7 in third inversion, but an interesting way to represent B7 altered harmony

on and on

There are a lot of useful options!

[JakeAcci]

two instances of seventh in the bass in a iim7b5 V7alt im7 vim7b5 context

first, compare conventional voicings:

X D Ab C F X

G X F B D# X

X C G Bb Eb X

A X G C Eb X

not better or worse, but certain different:

C X Ab D F X

B X A# D# G X

Bb X G Eb X C

X Eb G C X A

simply a different texture. If it were my solo gig, I'd be more happy to interpet the chords the second way.

C X Ab D F X = Dm7b5 drop 3 third inversion

B X A# Eb G X = G7#9#5 represented as Bmaj7#5 drop 3 root position

Bb X G Eb X C = Cm7 third inversion double drop 2 drop 3

X Eb G C X A = Am7b5 second inversion drop 2 and drop 3.

[NSJ]

NSJ, great questions.




If you are trying to make chord melody arrangements and are content to just play root position drop 2 or drop 3 chords and stick a melody note on top, then that's all you need. No need to bother with big spread voicings.

If you're not content to play that way, or to play in that context, then you look for other options, maybe things you find more interesting. I think many people find the process of discovering new vertical structures to be exciting and inspiring, and lead to new ways of playing, comping, composing, arranging, even playing lines.

. Another answer - he has to make use of these other voicings because he is often thinking of chord movement as individual line movement, and it doesn't always work out that the individual lines come together to result in familiar chord grabs.


Personally, I want to make solo improvisations that sound like a string quartet, so, well, I have some work to do...

I've been looking at Debussy and Ravel string quartet scores and there is MUCH use of these larger spread voicings. Beautiful stuff.

If you take one chord and go through all 24 inversions you'll find that each one has a very different character and feeling...personally, some speak to me more than others, some of them seem more part of 'my story' than others.

Another answer:

Each inversion provides a way to harmonize a melody note. In the context of a solo arrangement, you might simply find one of the other voicings to be a better sound than one of the ones you are more familiar with.

Another answer:

Arpeggiating these, or any inversion, can lead to more material for linear playing.

I could go on. I think one of the exciting things about this stuff is, like Goodrick says in the intro to the first Almanac book, is that different people will make different discoveries with the same raw material. I'm exploring this stuff personally to help with my solo guitar improvisations and to become more aware of multiple voice improvisation, but I think somebody else may get other benefits based on what he or she is working on or trying to do, or what sounds he or she is after. For example, Tim Miller has a huge chord vocabulary, he improvises over standard tunes and writes his own material for trio and solo guitar, and well, listen to his music...

What a well-articulated and thoughtful response. Thank you. I think what many of us play (the conventional drop 2s and 3s) is only because these are what we know, have practiced and internalized in a *meat and potatoes* sense. At the end of the day, we only play what we know, and what we know is only what we have internalized.

It's going to take a SHIT load of work to internalize all these possibilities. Even drop 2s and 3s, across the three possible stringsets, (6543, 5432, 4321, 6432, 5321) has a ton of permeations and combinations for each chord quality. I think it is well worth it. For all of the reasons you have laid out. But it reminds me that guitar and jazz is a lifelong endeavor.

I think that playing improvs that sound like a string quartet is about as lofty of a goal as one can imagine or set. First we have to internalize all these grips. Then we have to go beyond them. For myself, I can only imagine developing two lines moving on a guitar, at least in an improv sense.

[JakeAcci]

You're welcome!

Trying to internalize every inversion of every possible structure is probably impossible.

I think a better exercise might be to play around and search for ones you like, and internalize those - or let them inspire you to write or arrange something.

First we have to internalize all these grips.

We don't have to, necessarily. In fact, it might even be the wrong thing to do. I strive for an awareness of the voices when I am playing more than one note at a time, and I really try not to play like I'm just moving grip to grip. A lot of it is easier when I am familiar with more structures (hence, for me, the usefulness of this inversion list tool) but the process for me is to try to make each line sing. I find when I'm playing I often wind up playing chords that, vertically, I'm not familiar with at all - they are a result of how each line has moved.

[JakeAcci]

Thanks Jake. I love this kind of thing.

As for the 3 and 4 note permutations, I do the same thing, but I use Excel (spreadsheet). I put the 3 or 4 notes at the top, and then have links below for the permutations so that when I change the 3 or 4 notes at the top it changes the permutations below.

Cool! I use spreadsheet programs but don't know how to create the function you describe. Can you explain?

[edh]

Just a quick question, so I don't derail this thread. In the example that JakeAcci uses here:

"145 (sus4) from G" I add the F to this chord. It would now become a G7sus4 chord correct?

Thanks

[jsepguitar]

Cool! I use spreadsheet programs but don't know how to create the function you describe. Can you explain?

I just use =.

So for instance, if I was looking at Cmaj 7, I'd put C in cell A1, E in cell B1, G in cell C1, and B in cell D1.

Then if I had a row (let's say row 5) that gave me 1-5-7-3, the formula for cell A5 would be =A1, the formula for cell B5 would be = C1 (to give me the 5) etc.

So I have one row for each combination of 4 notes of which there are 24. Then anything I put in rows A1 - D1 will be permutated in the 24 rows below.

I also have a separate worksheet for 3 notes.

I tend to work in spreadsheets all day for my job, so it's what I'm most comfortable with.

[JakeAcci]

Just a quick question, so I don't derail this thread. In the example that JakeAcci uses here:

"145 (sus4) from G" I add the F to this chord. It would now become a G7sus4 chord correct?

Thanks

Yep, GCDF in G is 1 4 5 b7, G7sus4.

[NSJ]

I try to simplify everything, and this may sound silly. But, anyways. One thing I've been working on that dove tails with this excellent *cheat sheet* if you will, is to really get a grip on triads and spread triads, to easily facilitate the transition of one to the other. A more simple formula to transition from one form to the other form: start with the basic triad and its inversions and just move the middle note up an octave, so it becomes your top note.

ROOT: 135 becomes 153
1st INV: 351 becomes 315
2nd INV: 513 becomes 531

Just move the 2nd note up an octave. Boom. Easy piezy.

[jayv999]

For a very thorough and complete organization of all possible 4-note chord voicings see Ted Greene's V-System:
TedGreene.com - Teachings - V-System

This is beyond any other voicing system that I've ever seen.

[bondmorkret]

A cool idea. Probably not necessary for the way I see chords (I just visualise the intervals surrounding a chord tone and build a voicing from that), but a great idea. Nicely done

[JakeAcci]

Thanks bond...I agree, 'necessary' is definitely not a keyword here.

[OPherman47]

good stuff!!!, i wish i could find someone still selling the almanac... excellently put together post

[JakeAcci]

Thanks to Bako, Mick Goodrick, and Jon Damian it's now clear to me that all four note structures within a seven note scale fit into one of these five classifications:

1. 7th chord (1 3 5 7)

2. Quartal stack (1 4 7 3 when arranged in fourths, 1 3 4 7 in close)


3. Four scale tones in a row - cluster (1 2 3 4)

4. A triad plus its fourth (1 3 4 5). Goodrick refers to this as a triad over bass note - I don't like that terminology because once inverted, that classification makes no sense. From my perspective it makes more sense to classify it as a triad plus its fourth, and can have four possible bass notes just as any other four note structure.

5. A triad plus its second.

What's fascinating to me is that if you look at any whacky four note structure (anything in that cluster thread I started, for example) you see that they all fit into one of the five above.

Still fascinated by this as an organizational principle, but, specifically, triads over bass notes....

The only triads over bass notes that aren't just triads or seventh chords (C/E= triad, Em/C= seventh chord root position, C/B=Cmaj7 third inversion) are a triad over it's second or a triad over it's fourth. I already wrote that, but it's neat that from different chord scales we wind up with different possibilities:

triad over it's second:

C/Db
C/D
C/D#
Cm/Db
Cm/D

triad over it's fourth:
C/F
C/F#
Cm/F
Cm/F#
here's an interesting one: Cm/Fb...yes that's really a minor triad over it's major third, but structural it is consistent with the 'triad over it's fourth idea' if you look at it as being part of the third mode of Ab harmonic major: C Db Eb Fb G Ab Bb - triad is C Eb G, fourth is Fb. However...it's also just C/D# in inversion!

including augmented and diminished triads:
C+/D *
C+/D#
Cdim/Db
Cdim/D
C+/F# *
Cdim/F
Cdim/Fb (from Db melodic minor and harmonic minor)

*C+/Db and C+/F are just minmaj7 chords

Any other triad over bass note is just a triad inverted or a seventh chord.

The application of this perspective...

[bako]

Any other triad over bass note is just a triad inverted or a seventh chord.

Mathematically true but.......

C/Bb

C7 yes, but the story doesn't end there.

Most obvious is BbMa9#11

My instincts tell me (translate: as opposed to hours of concrete experience)
to place the chord in a larger context as a strategy to examine it closer.

Scale Context

This chord derives from the following scales:

F Major
F Melodic Minor
G Melodic Minor
F Harmonic Minor
F Harmonic Major
Ab Harmonic Major
C H/W Diminished

Can C/Bb function as extensions of any harmonies derived from these scales?

Bi-Tonal Chords might be another context to consider.

What other triads/chords is Bb a part of?

Are there any intriguing chordal sounds to be found?
Can C/Bb in any way represent these colors as an incomplete voicing?

[JakeAcci]

Bako, yes, I agree.

Earlier in this thread NSJ asked about third inversion seventh chords and I tackled it a bit:
https://www.jazzguitar.be/forum/compi...tml#post289285

Another significant thing is that a seventh in the bass of a 1357 chord means you actually have a triad on top of that bass note. Relative to the bass note, the triad is built form the second degree. Relative to the triad, the bass is the seventh degree.

So, you are dealing with triad over bass note harmony.

C/B is Cmaj7 in third inversion. But is also functions well as B7sus4b9 or as part of a B phrygian vamp, among other things.

C/Bb is C7 in third inversion. But it also can potentially be a voicing for Bbma7#11 or Bb7#11, or Bb lydian vamping (or lydian augmented or lydian dominant)

C+/B is Cma7#5 in third inversion, but that also works for B13sus4b9, second mode of A melodic minor

C+/Bb is C7#5 in third inversion, but a very common Bb9#11 voicing

Cm/B is Cminmaj7 in third inversion, but an interesting way to represent B7 altered harmony

on and on

There are a lot of useful options!

The reason I left out triad/7th in this case is the idea of inverting. C/Bb is only C/Bb when the C triad is on top and the Bb is a bass note. Any other inversion, it is just C7. But triad/4 or triad/2, like C/F, is a unique structure that doesn't turn into a familiar seventh chord once inverted.

I think triad/7 gets its on special category in a sense (or, well, in the context of what I've laid out in this thread) because they certainly do have a different sound...when you hear Am/G I don't think we all go "Am7," but once you invert it, the 'magic' is gone, unlike the triad/2 or 4.

[e_del]

Thanks to Bako, Mick Goodrick, and Jon Damian it's now clear to me that all four note structures within a seven note scale fit into one of these five classifications:

1. 7th chord (1 3 5 7)

2. Quartal stack (1 4 7 3 when arranged in fourths, 1 3 4 7 in close)


3. Four scale tones in a row - cluster (1 2 3 4)

4. A triad plus its fourth (1 3 4 5). Goodrick refers to this as a triad over bass note - I don't like that terminology because once inverted, that classification makes no sense. From my perspective it makes more sense to classify it as a triad plus its fourth, and can have four possible bass notes just as any other four note structure.

5. A triad plus its second.

Great, thank you!

This is a big hint, to organize the various harmonic possibilities.

Far from using them (speaking for myself) in the real world at the moment, but this draws a clear map of what to explore ... in the next decades!