Negative harmony is an idea that you'll sometimes see referenced in music theory. I think the term is due to Jacob Collier, and it's moderate closely related to the idea of undertone harmony or utonality. Let's look at it a bit.
First, we have a scale with a tonic and a dominant. If we draw a line between these two pitch classes on the circle of fifths, that's our guide for making substitutions in negative harmony - replace a pitch class with its mirror image across the line. If you're on a chromatic instrument, you can draw a line between the interval one minor third above the tonic and the interval one minor third below the dominant, and then substitute pitches by mirroring semitones across that line.
I don't believe in 12-TET interval space: G# isn't the same as Ab, the circle of fifths is actually a spiral of fifths, and a semitone doesn't fully specify a distance between pitches. So what can we do to fix the nonsense above?
Fix not just a tonic pitch class but a specific tonic pitch, {T} - i.e. decide on its octave. We could also specify a dominant pitch class {D} that's equal to {T + P5}. For any given note, find how far that note is under the tonic pitch and then add that distance on to the dominant pitch:
reflected_interval = (T - interval) + (T + P5)
This works fine whether the target interval is below the tonic or above or equal to it.
Let's work in C major for concreteness, and in rank-3 interval space, because we're adults. If we take the negative-harmonic-reflection of a C major scale with all natural pitch classes, we get the following scale out:
[G, F+, Eb, D+, C, Bb, Ab, G]
You can see that it's in reverse alphabetical order. That's a consequence of the reflection. These pitch classes are almost the same as the rank-3 spelling of the Eb major scale (or one its modes like C minor), but here the {F} pitch is raised by an acute unison where it would be natural in C minor. This means that negative harmony in rank-2 interval space is doing something similar to modal interchange, but in rank-3 interval space, it's doing something similar to modal interchange plus some mistuning.
Now, the key has changed from C major to approximately C minor, but we can see that the tonic of the reflected scale is G natural. Since G is the third scale degree in an Eb major scale, this means that we're instead in G Phrygian. Which is pretty cool.
If you take the negative-harmony-reflection of a chord, the pitches you get out are unambiguous, e.g. an F.maj7 spelled [F, A, C, E] goes to [D+, Bb, G, Eb]. I think there's a little ambiguity about how to name the chords though. The triads are easy:
F.maj -> G.m
[F, A, C] -> [D+, Bb, G]
But when you add on a 7th to the F.maj, should we keep calling the reflected version some kind of G.m chord, i.e. a G.mb13, or should we use the arguably more parsimonious name of Eb.maj7? I don't know. Whatever name we pick, and there might be more to choose from, if we start with a chord that's diatonic in C major, we're still going to get something that's approximately diatonic in C minor or G Phrygian, no matter what we name it. The problem with changing the tonic of the reflected chord as you add extensions in the original interval space is that sometimes your functional relationships from one side of the mirror don't translate exactly to other side, but usually they still translate fairly well: D.m and D.m7 reflect to Bb.maj and G.m7 respectively, and most chord grammars will treat these in similar ways within the key of Eb major / C minor / G phrygian.
One nice thing about negative harmony is that it let's you venture way outside of modal interchange: like, if you have a chord of B#.maj7 in a piece in the key of C major, it might not be immediately obvious how to translate that to a chord that has a similar relationship to the key of C minor, but in negative harmony you just turn the crank and get something out:
[B#, D##+, F##+, A##+] -> [Abb, Fbb, Dbb, Bbbb-]
which is to say
B#.maj7 -> Dbb.mb13 or Bbbb-.maj7
.
How do we use negative harmony? However you want. But I've got a few suggestions.
1) If you have a lead sheet with a melody and chords, keep the melody and reflect the chords. You melody won't generally have many tones in common with your reflected chords; this could be pretty spicy. When you reflect chords, the spacing of your voicing inverts: wide spacing in the base becomes wide spacing in the treble, and narrow spacing in the treble becomes narrow spacing in the bass. If you have block chords, this won't be a problem, but for chords vertical spacing that's uneven in interval space, I'd recommend revoicing your chords.
2) Reflect both your melody and your chords. If your melody is higher than your chords, it will now be under your chords. But we're revoicing our chords anyway, so after reflecting the melody, displace it by octaves as needed and revoice your chords below.
3) Replace just some of your chords with negative-harmonic substitutions. Use negative harmony as a chosen embellishment, not as a rote transformation.
How does negative harmony relate to utonality? Basically the utonal transformation of Partch inverts the frequency ratio space, while negative harmony flips the interval space. I think the negative harmonic reflection is more natural, but they're both fun, weird, and somewhat theoretically unmotivated tools for generating variations to play with.
In the chromatic conception of the negative-harmony reflection, our axis of reflection was between m3 and M3 over some tonic. What if instead of reflecting over the gap between two notes, we reflected on a single note?
Let's work in C major and reflect over C natural.
reflected_interval = T - interval
The C major scale reflected over C natural is
[C, Bb, Ab, G, F, Eb, Db, C]
This has four flats, so the key includes the rank-2 Ab major and C phrygian scales. Like with our previous definition of harmonic reflection, this produces a scale that is slightly mistuned in rank-3 interval space, since the rank-3 Ab major scale has its Bb lowered by an acute unison:
[Ab, Bb-, C, Db, Eb, F, G, Ab]
In this scheme, the melodies might reflect to sound like C Phrygian, but the reflected chords might not match a usual Phrygian composition. For example, reflecting a C.maj chord over C natural gives us an F.m chord in reflected space, so when we resolve our melodies on the tonic of C natural, we're usually not going to be playing a C.m chord, as you might have expected. Instead C.m is the reflection of F.maj, so your chords will sound most tonic when your original chord progressions were sounding subdominant. But if your melody had a chord tone in the original space, it will still have a chord tone in the reflected space (if you reflect both the melody and the chords), so things aren't that bad.
I like the original (Collier?) definition of negative harmony that changes all notes better than this one that reflects over the tonic and leaves one note unchanged, but it's entirely possible I'd prefer a variation like reflection over (tonic + P5) or something else, or reflection over a different gap, perhaps between (M2, m3), which splits the gap between P1 and P4 instead of P1 and P5. There's lots of room for experimentation here, since it's all unmotivated nonsense and none of the parameters are fixed by theory.
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