What is the correct process for deriving the 'mirror' or 'negative' harmony of a progression?

Question

I recently stumbled upon the concept of negative harmony in these threads:

• https://www.talkbass.com/threads/negative-harmony.533862/#post7227335
• http://www.jazzguitar.be/forum/theory/30394-negative-harmony.html
• http://decipheringmusictheory.com/?page_id=651
• http://m-base.com/essays/symmetrical-movement-concept/

I am trying to interpret it in the most basic ways, to get a feel for it.

The most useful post in the talkbass forum offers this info:

(progression I)

Cmaj7 (C E G B) E-7 (E G B D) G7 (G B D F)

(It's Mirror)

Dbmaj7 (Db F Ab C) F#-7 (F# A C# E) A-7(b5) (A C Eb G)

Notice how I got that? I started on the root of the first chord progression and spelled down new chords based upon the intervallic content of the first ones. C major 7 is C up a major third to E up a minor third to G up a major third to B. I took that same order of intervals, but used it going down instead of up. C down a major third is Ab, down a minor third is F down a major third is Db. This order of notes spelled a Db major 7 chord. You can do that with any quality of chord, and they will invert to some interesting things. Here's a basic list.

Major 7 => Major 7 a half step up (Cmaj7 to Dbmaj7)
Minor 7 => Minor 7 a whole step up (E-7 to F#-7)
Dominant 7 => Minor 7(b5) a whole step up (G7 to A-7(b5))
Minor7 (b5) => Dominant 7 a whole step up (C-7(b5) to D7)
Diminished 7 => Diminished 7 a minor third up (inverts to itself, essentially)
Minor major 7 => Augmented major 7 a half step up (C-(maj7) to Db+maj7)
Augmented major 7 => Minor major 7 a half step up

This seems straightforward enough. Each chord is mirrored around its own root note.

Part of the idea though, the way I heard about it, is that the negative chord will have an equivalent role. So for example a G7 tends to resolve to C. Therefore -G7, i.e. A-7(b5), should equally resolve to C.

But then I started to wonder if this meant the tones of chords in a progression should be mirrored around the key of the piece, rather than the root of each chord?

That would generate different results from above. Can anyone shed any light?

Answer 1

"Negative harmony" is a term from the theorist Ernst Levy and his book A Theory of Harmony. Here's what's actually happening:

Let's say you're in C. The idea is that the "axis" of C is the perfect fifth C/G. So, when you have a G7, you're actually inverting it around this C/G axis. Or, put another way, the halfway point between C and G is right between E and E♭, so you're actually rotating it around that point.

As such, G becomes C, B becomes A♭, D becomes F, and F becomes D. So in terms of negative harmony, the equivalent of G7 is Dhalfdim7 (or Dm7♭5).

The idea is that the voice-leading tendencies are just inverted. The leading-tone B, which wants to go up by half step, is now A♭, which wants to go down by half step.

What makes this confusing is that it's close to but not the same as other ideas. It's really an evolution of a prior concept (typically connected with 19th-century German thought) called harmonic dualism. From a dualist standpoint, the chords will be inverted around their tonic; thus C major (C E G) inverts around its tonic to become "C dual minor" (F A♭ C). Yes, you read that correctly; harmonic dualists name minor triads by what we consider their fifth!

This is also not to be confused with the twentieth-century notion of pitch (or pitch-class) inversion, where music is simply rotated around a given axis.

Answer 2

In addition to Richard's excellent answer, which I've accepted, I just wanted to highlight some other approaches to this question I found on the internet.

One problem in investigating this subject, as a casual musician rather than music theory major, is a) not having the Levy book b) not sure if I'd understand it anyway and c) of the few internet sources discussing the topic that aren't way over my head, hard to judge if they're getting it 'right'.

So for point c) I'm going to assume that musical prodigy, Jacob Collier, having apparently discussed it with Herbie Hancock (!) knows how this is supposed to work. With that in mind there's some clues in this video:

He talks about using a C/G "axis" for the key of C, and thus transforming G7 into -G7 = Fm6 (aka Dø). Furthermore he transforms a VI-II-V-I chord sequence in C (A7 D7 G7 C) into E♭m6 B♭m6 Fm6 C. (I think the C itself is not transformed here).

So with this in mind let's investigate...

1.

Firstly there is this website:
https://personalprojectnegativeharmony.wordpress.com/

They don't talk about the axis at all. Instead they use the idea that the 'negative scale' of C Ionian is G Phrygian.

They also talk about other elements of Levy's theory (I assume it comes from Levy, though it may be saxophonist Steve Coleman's interpretation of Levy) such as a 'generator' as well as the tonic.

I'm still not sure what is the 'generator' but my attempt to follow their method is as follows:

• intervals of G7 chord with respect to C major are 5-7-2-4
• count 'downwards' in the G Phrygian (i.e. 'negative C major') scale, starting at G
• a five scale steps down from G is C
• a seven scale steps from G is A♭
• a two scale steps down from G is F
• a four scale steps down from G is D
• C-A♭-F-D = Fm6

So far so good.

We can try also to generate 'negative A7' (our VI chord). The intervals with respect to C major are 6 ♭2 3 5. Jacob says this should be an E♭m6 = B♭ G♭ E♭ C.

• count down six scale steps from G to get B♭
• count down two steps from G to get F ...but since it's supposed to be a ♭2 interval and I'm counting down I should sharpen it to G♭ ... ok
• count down three scale steps from G to get E♭
• count down five steps from G to get C

OK this works too... but I don't think this is how they describe the method in the article!

"As Steve Coleman shows above, the negative version of the C-E-G triad is the minor triad F-Ab-C with C being the generator and the tonic. Remembering that the negative version of the C major scale is Gm phrygian, going downwards from G, we can find the F minor triad in that scale."

So what they describe here is to count down 1-3-5 in G Phrygian starting at C rather than G.

EDIT: I think this is a mistake in the blog, perhaps misinterpreting what Steve Coleman was trying to explain here, and that's why I've been so confused. The text says to count down from G but somehow to arrive at an F Minor triad. The negative version of C Major triad should be a C Minor triad C E♭ G. We can arrive at that by counting down 1-3-5 in G Phrygian starting at G as they suggest. The F Minor triad F-A♭-C must be -G in the same way that Fm6 is -G7 (i.e. built off the fifth degree of the negative scale) in order to be consistent with everything discussed elsewhere.

Then they continue:

Now remembering also that we must look at this Fm triad as actually being generated from C and going downwards, the next degree of the Gm Phrygian scale, a third down from F, ie the “7th”, would be D.

So 1-3-5-7 (natural 7th) counting down from C gives us the Fm6...

This is why the negative dominant of chord I of a key is a minor sixth chord.

Ok but why did we count 1-3-5-7 to get -G7 for the key of C?

I have to conclude I don't understand how to use the theory as described on that website.

However it might be a purer version of the theory than anything discussed here. It also has some interesting discussion re Steve Coleman's efforts to relate negative harmony to jazz, and why they prefer to call it Fm6 rather than D half diminished.

EDIT: after re-considering, reading other sources, and correcting the mistake above I think I can answer the question I had; "why did we count 1-3-5-7 (starting at C) to get -G7 for the key of C?"

If we do it first in positive land we find that counting 1-3-5-7 in C Major scale starting at G (the fifth degree) will give us the notes for G7. In negative land the scale is G Phrygian backwards and the fifth degree is C. Counting 1-3-5-7 in that scale, from C, gives us the notes of Fm6.

(Also: it is basically a coincidence that the intervals of C Major scale counting 1-3-5-7 from G to spell G7 are similar to the intervals in the G7 chord relative to its own root, 1-3-5-♭7. This kind of confused me a bit previously. The point is to come up with a way of relating the notes in the chord as intervals in the parent scale, then count the same intervals in the negative scale.)

2.

Going back to the YouTube clip, there was a useful suggestion in a comment from user Huw Price.

He says we can use the Circle of 5ths to get our negative harmony.

Let's try it out. Assuming we're in the key of C we want to fold our circle in half like so:

We could almost justify calling this a 'C-G axis'

Now we have all 12 notes, and to get the negative of each just find its reflection on the other side of the circle, i.e. C -> G, F -> D etc.

We can find 'negative G7' by reflecting the notes G B D F -> C A♭ F D

We can get Jacob's 'negative VI-II-V-I':

• -A7 = B♭ G♭ E♭ C = (E♭m6)
• -D7 = F D♭ B♭ G = (B♭m6)
• -G7 = C a♭ F D = (Fm6)

This seems like an easy and successful method.

And for another key you'd just draw your axis between the root and fifth and get your reflections that way.

3.

Perhaps because I had a guitar in my hand, instead of a pen and paper, I'd initially found it difficult to see how to apply the method from Richard's answer.

After exploring the Circle of 5ths way, it seemed like this would also be easily done visually.

Write out the twelve notes in a row, starting from the ♭7 of your key and finishing with the 6.

e.g. for C:

In very much the same way as the Circle of 5ths, we can get the negative of each note by picking its opposite from across the reflection line between E--E♭.

4.

Here's some further handy 'shortcut' info, again taken from Huw Price's comments on the YouTube video.

Firstly, let's take the 'chord scale' of C major:

C Dm7 Em7 FΔ7 G7 Am7 BØ

The negative harmony counterparts of these are:

• A♭Δ7 (CEGB --> G E♭ C A♭) (taking Cmaj7 as the starting point)
• Gm7 (DFAC --> F D B♭ G)`
• Fm7 (EGBD --> E♭ C A♭ F)
• E♭Δ7 (FACE --> D B♭ G E♭)
• Dø (GBDF --> C A♭ F D) aka Fm6
• Cm7 (ACEG --> B♭ G E♭ C)
• B♭7 (BDFA --> A♭ F D B♭)

Huw points out that these form an E♭Δ chord scale.

We can write the negative transformations in general terms as:

• IΔ7 --> ♭VI Δ7
• IIm7 --> Vm 7
• IIIm7 --> IV m7
• IVΔ7 --> ♭III Δ7
• V7 --> II ø ...aka... IV m6
• Vim7 --> I m7
• VIIø --> ♭VII 7

5.

From Adam Spiers's comments, this is another source we can look at:

with accompanying PDF:
http://www.mediafire.com/file/qntuqa7ifa6ex5z/Negative_Harmony_%232+-+Cadences+and+Voice+Leading_Fiorini_Marco.pdf

This basically explains Method 1 but also relates it to the ideas of Barry Harris... i.e. a relationship between diminished chords, dominant 7ths and minor 6ths.

So far in this post I have been referring to eg C -> -C = Cm, G7 -> -G7 = Fm6. And if the positive scale is C Major then "negative C Major" is G Phrygian backwards.

However in the PDF above the author uses a different convention, so you have pairs like C | -C (Cm), G7 | *-C7* (Fm6) and for the scales "C Major" and "negative G Major" (which is still G Phrygian backwards).

So in this scheme the negative chords and scales have the real (i.e. positive) name of the note from the negative scale they are built from as their root note. (i.e. what I called -G7 and they call -C7 is built starting from C, the fifth degree of the negative scale).

But this seems to me more confusing as you have to do an extra mental step to relate it back to the positive key. Also why only translate the root note of the name in this way and not rest of the chord? i.e. -C7 is not a 7th chord in real (positive) terms... the notes of Fm6 with a root of C spells, er, I don't know how you'd call it. Whereas -G7 has no G in it but at least tells you clearly the chord it was generated from.

I have no idea what the correct convention is for naming the negative chords but -G7 = Fm6 so far seems to make the most sense to me.

However one thing to note is with this scheme the negative names only make sense in relation to a positive key. For example in the key of C we have -G7 = Fm6. If we transpose down to B♭ then we will have -G7 = D♭m6 (G7 being now in the VI chord position).

Whereas, according to the method in the PDF, if we have -C7 = Fm6 in C and we transpose up to D then we still have -C7 = Fm6, where this represents the negative B7 (the VI chord again). Or if we transpose to G then we have -C7 = Fm6 again, this time as the negative of A7 (the II chord).

So this stable naming seems an argument in favour of the convention used in the PDF above and is maybe more correct. I would welcome further comments on this.

Answer 3

I think another way to look at it is to think of the intervals of each note with respect to E and then apply those same intervals from Eb and there would form each negative chord.

Answer 4

As Collier points out the axis is between the minor and major 3rd of a key, E and Eb in the key of C. That means you simply count an iterval upwards from E (or the major third of your key) and then reflect it downwards from Eb (or the minor third of your key).

A simple way is to start with the chord tone closest to the axis and calculate the relative intervals from there. In the case of A7 you would start with E (right on the positive axis), then up a minor 3rd to G, major 2nd to A and major 3rd to C#. E-G-A-C#=A7

The reflection starts on Eb (right on the negative axis), down a minor 3rd to C, down a mjor 2nd to Bb and down a major 3rd to Gb. Eb-C-Bb-Gb= Ebm6.

G7 starts on F (1/2 step up from E) etc. Its refletion starts on D (1/2 step down from Eb) etc.