A chart of the common chords of tonal harmony and their negative harmony mirrors. This isn’t meant to provide an introduction to negative harmony (there are already great resources on that), but instead to provide a reference chart for composers trying to incorporate negative harmonic concepts into their music. It’s also meant to serve as a prequel for upcoming posts on negative mirrors of common chord progressions, turnarounds, and jazz forms.
It’s possible to mirror chords across a number of axes, but for this and upcoming posts I’m specifically looking at the b3/3 axis. This is the axis Jacob Collier talks about as “converting perfect to plagal” and maintaining equivalent “tonal gravity” between the original and mirror chords.
My specific process for deriving these mirrors was to flip each note of the original chord across the b3/3 axis, then select the root note by reflecting the original root across the tonic (1) axis. Part 3 of Jazzmodes’ negative harmony series has some more explanation on why it makes sense to select the root this way. In short, it’s because this will cause the mirror roots to always move proportionately to and opposite of the original roots on the circle of fifths (descending fifths become ascending fifths, etc).
Note that in chords without a perfect fifth above the root, that method actually gives a root note that isn’t in the chord. In those cases the mirror root is undefined/ambiguous, so I just picked a voicing that made sense for the given mirror pitch class set.
** ambiguous mirror root
Alternate Method for Defining Mirror Chords
Another possible way you could define a chord mirror is by reflecting the entire chord-scale across the b3/3 axis and then constructing a chord using that chord-scale built on the mirror root. The mirror triads would be identical, but the extensions would change. For example, Imaj7 would reflect to i-7 instead of i-b6.
This might be more palatable because it uses familiar triadic harmony, but it would also change the harmonic gravity between the original chord and its mirror. For example, V7 and its mirror, iv-6, have the same harmonic gravity in their leading tones (7->1 reflects to b6->5 and 4->3 reflects to 2->b3). If you instead used iv-7, the voice leading would change and no longer perfectly mirror the original.
How to Mirror a Chord-Scale
A chord-scale can be mirrored the same as a regular chord. Select the mirror root by reflecting the original root across the tonic axis. Then find the rest of the notes by reflecting the entire pitch-class set across the b3/3 axis.
We can make use of a shortcut here using the above table because a given quality of chord-scale will always reflect into the related quality of mirrored chord-scale. This relationship is described in the table below:
|Original Chord-scale||Mirror Chord-scale|
Using that, any chord-scale can be mirrored by reflecting the root across the tonic axis and using the mirror quality relationship from the table. For example, a bIIImaj7 (lydian chord-scale rooted on b3) would reflect into a vi-7 (phrygian chord-scale rooted on 6) because b3 reflects across the tonic to 6 and lydian reflects to phrygian.
As with chords, this depends on the chord-scale having a perfect fifth above the root. If there isn’t a perfect fifth above root, the resulting mirror root is ambiguous and there are multiple ways to interpret the mirrored pitch-class set.
I’m not going to write a whole chart of these because they’re fairly easy to derive from the main chord chart. Also, not all of the chords listed have a single unambiguous chord-scale; some of them are dependent on the context in which a chord is being used and/or stylistic considerations and that goes beyond the scope of this post. Here are just a few to get started with:
|Chord||Chord-Scale||Mirror Chord-Scale||Mirror Root and Possible Chord|
|vii°7||Whole-half diminished||Half-whole diminished||vii°7|
References and further reading:
June Lee “Interview: Jacob Collier (Part 2)”
Jazzmodes “Negative harmony’s mirror world”
Rick Beato “Musical Palindromes & Negative Harmony (what?)” (Note: he derives mirrors by reflecting everything across the tonic axis, which is why we end up with different modal pairs.)