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?