Negative harmony part 2

By now I thought I would know exactly what this negative harmony thing is, to the point where I could explain it clearly in a paragraph or two, and perhaps apply some of it to my own humble efforts to play jazz piano. I was hinting as much in my previous post, which you might want to read before this one.
No such luck. It turns out there’s a swirling vortex of ideas out there, a lot of vague excitement, little solid information and (consider this a warning sign) very little actual music.
Somewhere at the centre of all this is the unifying idea of reflecting musical objects in the vertical dimension. Simple enough. Applied to melody, this is the idea of inversion, or turning a tune upside down, as practised by everyone from J.S. Bach to Anton Webern.
Bach generally went for diatonic inversion, with tune and inversion moving by equal and opposite numbers of scale steps, either in the same key or in closely-related keys. The results, though often wonderful, don’t sound so out-there as some of the things jazzers are doing today. We need to channel Webern rather than Bach, and reflect melodies chromatically. That means replacing each upward interval by a downward interval of the exact same number of semitones.
A tone-row used by Anton Webern in Variations for Piano, opus 27. The 12-note sequence can be inverted in pitch and in time, generating four versions altogether.¹
We can invert melodies, scales, chords, or whole harmonic progressions chromatically.
For scales, chords and progressions, a good way to picture chromatic inversion is to plot the relevant notes on a circle of twelve ‘pitch classes’ and watch what happens when each note is reflected in a chosen axis. The chromatic circle allows us to analyse harmony at an abstract level, ignoring octaves and the details of voicing. The pitch class C stands for all the Cs on the keyboard.
Most chords change in quality when they are inverted. Major and minor triads are each other’s mirror images. This creates a characteristic sound in negative harmony.
chromatic circle Cmaj Cmin
The axis chosen here (diagonal line) reflects a C major triad into a C minor triad.
Four-note chords are more problematic: the chord’s spelling, its harmonic function and the way we hear it may change, depending on which tone we identify as the root. In the following example, a dominant chord (D7) reflects into a chord which could be spelled as A-7b5 or as C-6.
D7 ambig
Here, A-7 reflects into D-7, or is it F6?
A-7 ambig
There is a rule for this. Reflect the root, and then lower it by a perfect fifth. In our example, the root of D-7 is D, which reflects into G. Drop a perfect fifth to C, and make that the new root. The reflected chord should therefore be spelled as C6 rather than A-7. Note that if you now reflect the reflected chord, the root C goes to A. Lower that by a fifth and we are back to D, the original root. The process is reversible.
In the other example, G7 goes to G-6 and, reflected again, back to G7.
The best case for adopting this rule is made by the ear. I have only heard a few negative harmony examples, but already I prefer those which follow the rule to those which ignore it. You will inevitably hear a theoretical justification based on the theories of Ernst Levy, but I am not sure this is necessary. I am confident that conventional harmonic theory could account for the rule if necessary.
Here is how I am thinking about it until something better comes along. Adding sevenths to a sequence of triads does not change the root movement, nor should it affect the root movement in the progression’s negative version. This has two welcome consequences:
  • the pattern of major and minor, light and shade in the tune will be reversed, but structurally unaltered
  • common patterns of root movement, such as 36251, reflect into patterns which have a similar logic; movement in descending fifths, for example, reflects into rising fifths
That’s it, apart from minor tweaks for diminished and semi-diminished chords.
Jacob Collier, in June Lee’s now-famous video, talks mainly about inverting chords and progressions. He demonstrates how root movement in descending fifths (think Autumn Leaves, Fly Me To The Moon, rhythm changes) is transformed into a chain of rising fifths or plagal cadences.
I think Collier is suggesting a new palette of colours for composers starting from a blank sheet. But the jazz world has shown more interest in negative harmony as a reharmonisation technique.
Pianist Phil Wilkinson has done a nice negative-harmony version of John Coltrane’s Giant Steps, which is on his new trio CD. This is a demo version:
The original tune contains many V-I and ii-V-I progressions which, in this version, are replaced by tasty-sounding plagal cadences. The destination chords remain the same, so the tune keeps its essential bone structure, with the harmony rising and falling by major thirds.





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Let’s look at how the G major cadence, which begins at bar 4, is affected. The reharmonised cadence, D-6, C-6, G (v6-iv6-I) is the negative equivalent of the IV7-V7-I cadence C7, D7, G. You can see the relationship between the positive and negative versions by ‘flipping’ the first two chords around an axis between Bb and B on the keyboard. The G major chord is not flipped: if it had been, it would have come out as G minor. Note that Phil has not simply applied a formula to the original ii-V7-I cadence. He has picked some fresh colours from the negative harmony palette.
The tune moves through three keys – Eb, G and B – and in each case the corresponding chords are taken from the negative palette of that key. This means that three different axes are used in the course of the piece. Musically, it works. Phil achieves a convincing reharmonisation of the head, and goes on to improvise fluently over the new changes.
Gmaj Gmin axis
The axis chosen here reflects a major key (G) into its parallel minor. Like our first example, it follows the ‘Jacob Collier rule’. Coloured dots indicate scale tones.
Let’s take a closer look at how this works in, say, G major. Following Jacob Collier’s formula and using an axis between Bb and B, the notes of the G major scale, G A B C D E F#, are flipped into the following: D C Bb A G F Eb. These are the notes of G minor (aeolian).  Consequently any diatonic chord of G major will reflect into a diatonic chord of G minor.
A nice feature of this choice of axis is that if you apply the negative harmony selectively, perhaps to sequences as short as a single chord, you won’t end up in some crazy unrelated key. The music will simply shuffle between the parallel major and minor, which is something a lot of tunes do already.
You can also flip non-diatonic chords such as secondary dominants, diminished sevenths or, I guess (though it’s unexplored territory for me) dominants with altered extensions. Just as diatonic chords in the major flip into diatonic chords in the minor, non-diatonic chords will always flip into non-diatonic chords².
Other choices of axis will reflect scales and chords in a similar way, but landing in different keys. Reflecting G major on an A/Eb axis gives E minor, the relative minor. This might offer another way to apply negative harmony selectively within a piece, while maintaining harmonic continuity. The harmony would shift between relative major and minor, which are, obviously, highly compatible keys.
Gmaj Emin axis
Worth investigating? Departing from the ‘Jacob Collier rule’, this axis reflects G major into its relative minor. The set of scale tones  is unchanged by reflection but the G major triad reflects into an E minor triad.
If everything in a piece is inverted – melody, chords, bassline, from beginning to end – there are no compatibility issues, and all axes are good. Just choose a key, any key, for the negative-harmony version, and pick an axis that will land you there. You’ve got a choice of twelve keys and twelve axes.
It is clear that I will need another post, or possibly two, to do justice to this subject. I still want to examine the improvisation methods that saxophonist Steve Coleman has developed over his career, and explore the mystical theories of the Swiss composer Ernst Levy, who is often named as the originator of negative harmony.
To sum up:
  • Negative harmony suggests new, or less familiar, harmonic progressions for composers
  • For performers and arrangers, it offers new ways to interpret or reharmonise existing tunes
  • Chord changes can be wholly or partly replaced by their negative equivalents
  • Melodies can also be inverted
  • The most popular method reflects a major key into its parallel minor; this is useful for selective reharmonisation
  • Music can be reflected into other keys, such as the relative minor, by changing the ‘axis’; the possibilities seem to be largely unexplored
  • You need to get the root movement right.

1. The original uploader was Hyacinth at English Wikipedia – Created by Hyacinth (talk) 06:45, 7 December 2010 using Sibelius 5, Public Domain,

2. In the minor key I am using ‘diatonic’ to mean ‘using only the notes of the aeolian scale’. I know that is controversial. I will deal with this in a future blog post.

3 thoughts on “Negative harmony part 2”

  1. I’m not quite clear on what you mean by the “Jacob Collier rule”– does that describe simply placing the axis in between pitch classes as opposed to directly on them?


  2. I meant placing the axis between the minor and major third, as described by Jacob Collier in the famous video. I don’t think he meant to rule out other possibilities, such as placing the axis on or between other pitch classes. That’s just the one that caught his attention.


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