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The square root of negative harmony is imaginary harmony.

Posted by u/damien_maymdien on r/musictheory

Original post via Teddit

Negative harmony, for anyone out of the loop, is a process built from the recognition that if you use the intervals of a major triad (major third followed by a minor third), but go down instead of up, you get a minor triad, and likewise a major scale turns into a minor scale. Because this is basically just a reflection, it also works in reverse—minor turns to major just as major turns to minor. It usually refers to the specific version of this reflection of intervals that turns, say, a C major chord into a C minor chord when you’re thinking in the key of C. That involves a reflection of each note about the point midway in between E and E♭ (or equivalently about the point midway in between A and B♭). That results in the 12 notes going to the following new notes in negative harmony:

C  C♯ D  E♭ E  F  F♯ G  A♭ A  B♭ B  
G  F♯ F  E  E♭ D  C♯ C  B  B♭ A  A♭

Besides sending C major to C minor, it also sends G7 to Fm6=Dm7♭5=~The Christmas Chord~, which is one compelling explanation for why iv->I sounds good a lot of the time—it’s just substituting the dominant chord with the “negative” dominant chord.

But the name “negative harmony” fits this transformation well from a *mathematical* perspective, not just in the musical sense of major = positive and minor=negative, or even intervals being stacked in the negative direction.

Just like -1^2 = 1, applying the negative harmony transformation twice brings all notes back to themselves. C goes to G, but G goes back to C when you apply it a second time. In other words, the negative harmony transformation is a square root of the identity transformation (the “transformation” that just maps every note to itself).

But we can go further.*

In math, the square root of -1 is the imaginary number i. So analogously we can take the square root of the negative harmony transformation to get a transformation defining Imaginary Harmony.

There are actually 120 different transformations that, when applied twice, equal the transformation to negative harmony. But we can decide between them based on particularly appealing symmetries and musical utility, just like negative harmony stands out from the other 10,394 transformations that equal to the identity transformation when applied twice.

The transformation I’m suggesting as the canonical “imaginary harmony” transformation is the following:

C  C♯ D  E♭ E  F  F♯ G  A♭ A  B♭ B  
F♯ C  B  B♭ A  A♭ G  C♯ D  E♭ E  F

This has an appealing symmetry in the way it’s two groups of 6 notes in the transformed scale that move chromatically, but more importantly, it has cool musical properties: it transforms the C major scale to the F♯ melodic minor scale (and since it’s the square root of negative harmony, it transforms the F♯ melodic minor scale to C minor). It also transforms G7 to C♯7=D♭7, providing a motivation for the tritone substitution just as negative harmony provides a motivation for the G7->Fm6 substitution.

Just like -i is an additional square root of -1, doing the imaginary harmony and the negative harmony transformation subsequently (or equivalently doing the imaginary harmony transformation three times) gives another square root of negative harmony. This “negative imaginary harmony” maps the C major scale to F♯ melodic major, making the cycle C major -> F♯ melodic minor -> C minor -> F♯ melodic major -> C major, etcetera.

I’d be curious if anyone else can suggest an alternate transformation that’s a square root of negative harmony with different compelling musical features that I might have overlooked. If you know a little abstract algebra, the simplest way to to consider all of the different 120 square root possibilities is to write the transformations in cycle notation, i.e. negative harmony = (CG)(FD)(B♭A)(E♭E)(A♭B)(C♯F♯)

If you want to get real spooky with your harmony, there are also cube roots and 6th roots of the negative harmony transformation. But I’ll leave those as an exercise for the reader.