Are any keyboard methods or music theory concepts based on keyboard mirror symmetry?

Question

About a year ago I noticed there is a kind of mirror symmetry on the piano keyboard regarding the shape/distance of the black and white keys when starting at C major and then chromatically descending for the minor keys and chromatically ascending for the major keys.

I tried to make this chart to demonstrate...

MINOR                     │ MAJOR
                          │
key sig. key. piano triad │ piano triad. key. key sig.
                          │
2 #      Bm   w w B       │ B w w        Bb   2 b
5 b      Bbm  B B w       │ w B B        B    5 #
0        Am   w w w       │ w w w        C    0

etc...

The w's and B's mean white or black piano keys.

Each ‘pair’ has the triads as a sort of inversion of the other. The count of accidentals in the key signatures is the same, but the sharps/flats are flipped.

I also noticed an inverted or retrograde symmetry regarding the scales on these tonics.

The retrograde of the 5th mode of one scale produces the sequence of the 'symmetrical' matching scale, ex...

Bb major: B w w B w w w
                  |
          b minor w B w w B w w (retrograde of the 5th mode of Bb)

I figured this out when trying to compare the fingering issues betwwen left and right hands and realizing that, for example, B minor is the left hand isn't really like B minor in the right hand. Rather B minor is like B♭ major in the right hand. At least it is in terms of shape.

I applied this to 5 finger position for stuff like little scale turns shakes in thirds, etc.

Are there any piano methods based on this kind of symmetry?

The only theory concept I know which seems related is Riemann ‘dualism’ with the scale inversion.

Are there other related theory ideas? Something like a harmonic sequence or modulations?

Answer 1

German pianist and composer Victor Alcántara has studied the subject of the keyboard's mirror symmetry in depth; he has published a book ("Palindromorphy") on this.

Answer 2

Some melodies are the "inversion" of another melody about an axis which need not be contained within the melody or starting key. Your example could be used to invert a melody around the C axis.

Another similar notion occurs in the "omnibus progression" (AKA wedge progressions.)

Answer 3

My observations on this topic:

1. (Applicable to all instruments) If we divide a major scale 7 notes by the middle note(4th note), the left three notes will have similar intervals that of right three notes.

Example: In C Major,

Left Side

C to D is Whole and D to E is Whole

Right Side

G to A is Whole and A to B is Whole

2. (Applicable to all instruments) If we divide a minor scale 7 notes by the middle note, the left three notes will have the mirror intervals that of right three notes.

Example: In Bb minor,

Left Side

Bb to C is Whole and C to Db is Half

Right Side

F to Gb is Half and Gb to Ab is Whole

3. (Applicable to Piano) If we divide a major/minor scale 7 notes by the middle note, the left three notes will have the same number of black and white keys that of right three notes, In case of odd number of white/black keys, left side will get extra black and right side will get extra white for key signatures with b while right side will get extra black and left side will get extra white key for key signatures with #.

Example: In C Major,

Left Side

 C,D,E are White, 3 white and 0 black

Right Side

 G,A,B - 3 white and 0 black

Example: In Bb minor,

Left Side

 Bb,C,Db - 2 black, 1 white 

Right Side

 F,Gb,Ab - 2 black, 1 white 

Example: In Bb Major, (5 bs)

Left Side

 Bb,C,D - 2 white, 1 black (extra black in left side)

Right Side

 F,G,A - 3 white, 0 black (extra white in right side)

Example: In A Major, (3 #s)

Left Side

 A,B,C# - 2 white, 1 black (extra white in left side)

Right Side

 E,F#,G# - 1 white, 2 black (extra black in right side)

Note: Also when we play melodies in these left and right sections in same rhythm consequently, we will get some kind of call and response melody feelings.