Music theory on a grid: negative harmony

I felt a little bit fullstack, so I made a playable web music keyboard. It works (unless you use a smartphone or tablet to read this).

Let's use a grid to explore the concept of inversion.

This is a major third from A pointing forwards:

.   .   A   .   C#  .

This is a major third from A pointing backwards:

F   .   A   .   .   .

In music theory, flipping all interval from a given note is called inversion.

How do you inverse an interval that doesn't lie on a row? You flip both axis.

This is the A chord:

.   .   .   E   .   .
  .   .   A   .   C#
.   .   .   .   .   .

This is the Dm chord: all intervals from A are flipped.

.   .   .   .   .   .
  F   .   A   .   .
.   .   D   .   .   .

This is F major pentatonic scale:

.   C   D   .   .   .
  F   G   A   .   .
.   .   .   .   .   .

This is the same scale inversed from A:

.   .   .   .   .   .
  .   .   A   B   C#
.   .   .   E   F#  .

Now negative harmony. There's so many talks about this topic, all full of confusion. Honestly, I didn't even try to read any of them. Instead I went on the internet and found this pic: https://i.imgur.com/tMNyfze.jpg

So “negative harmony” is simply an “inversion around an axis”. And “inversion around an axis” is actually a composition of inversion and transposition (indeed, the proof is trivial).

How do you do an inversion that transforms A into E? You it in two steps. First, you inverse from A. For example,

.   .   .   E   .   .
  .   .   A   .   C#
.   .   .   .   .   .

becomes

.   .   .   .   .   .
  F   .   A   .   .
.   .   D   .   .   .

Then, you apply a transposition that moves A into E. So

.   .   .   .   .   .
  F   .   A   .   .
.   .   D   .   .   .

becomes

.   C   .   E   .   .
  .   .   A   .   .
.   .   .   .   .   .

Now note one interesting thing. We started A major chord. After applying A-E inversion, we have got A minor. So inversion along the dominant arc turns major into minor and vice versa, like in the harmonic dualism!

I do not believe in dualism, but insn't it kinda cool?

By the way, do don't have to restrict yourself with only dominant arc. Why not invert along a major third arc?

.   .   .   .   .   .
  .   .   A   .   C#
.   .   .   .   F#  .

So A major becomes F# minor.

PS: the essential property of inversion is preserving interval relations between notes. This includes interval spectrum as well as step relations.

UPDATE: the grid slightly differs from the one that was in the previous post. In the previous version, the fifth looked like

  .   .   A   .   . 
.   .   .   E   .   .

In the current version, it looks like:

.   .   .   E   .   .
  .   .   A   .   . 

By the way, in both versions, an octave look like:

.   .   .   E   .   .
  .   .   .   .   . 
.   .   .   E   .   .

So grids are essentially the same when you only consider notes, but are different when you also consider voicing. By the way, this is how chord inversions look on the grid:

Root:

.   .   .   .   .   .
  .   G   .   .   .
.   C   .   E   .   .

First:

.   C   .   .   .   .
  .   G   .   .   .
.   .   .   E   .   .

Second:

.   C   .   E   .   .
  .   G   .   .   .
.   .   .   .   .   .

The perfect fifth direction was changed so you can easily play the grids you see in the web app.