When looking at Barry Harris Half Diminished Scales in music theory, we see he adds a b6 to normal scales - both Major and Minor resulting with8 notes per scale.

🎵 Standard Major Scale Vs. Barry Harris

T = Tone = 1
S = Semitone = 0

Standard Major

Standard Harmonic Minor

Standard Melodic Minor

Barry Harris - 8 Note Dim 6 Major

Barry Harris - 8 Note Dim 6 Minor

Symmetry of Scales

Let's break the scales in half - essentially 2 bytes

Major 6 Dim:

NOTE: Series (ii) is the direct inverse of series one.
You'll notice that as we use this method, as long as you make series 2 the direct inverse of series 1 (ie. 1 becomes 0 and 0 becomes 1), you will always get a symmetrical harmonically sound scale that:

1. Works harmonically
2. Fits in to 1 octave perfectly
3. Derives the Harmonic Binary contrast between Root Chord vs. Position 2, Dim Chord

Example:

Looking at the chords this produces, as Barry Harris teaches, you only really have 2 chords if you play alternate notes together.  In the key of C:

Note that there are only really 3 Diminshed groups.  I call these X, Y and Z just for the sake of labeling them.  When I'm working in a key, feeling whether I'm in an 'X', 'Y' or 'Z' key allows me to play and improvise appropriately using what Barry Harris calls 'Playing with your brothers and sisters'.  That means that if you drop the root note of any diminished chord (the first note if build up in 3rds), you will get a Dominant 7th chord.

The Standard Diminshed Chords:

So using Column 'X' as an Example:
Drop 'D' a semitone from D F Ab B = Db F Ab B = Db7

Barry Harris' 'Brothers and Sisters' are therefore the group of 4 Dominat chords that are created by dropping each one of the 4 notes of columns X, Y or Z.  These chords and any scales or movements based on them can be interchanged with their 'Brothers and Sisters'.

Minor 6 Dim (Barry Harris)

Note - this is where I posit where Barry Harris uses the same 'ii' section (0010) for both Maj6 and Min6, I posit that there is another way of looking at this to come to a slightly different (and same) result depending on whether you're doing uni-directional scales vs. contrary 'Positive' vs. 'Negative' harmonic scales. Harmonic scale here not as in standard western music, rather following the harmonic series in a positive or negative direction.

Example:

Look at the Binary 'bytes' of i vs. ii:

If you look at it the same way we did for major, it should be:

Which would generate (in C):

Which is ACTUALLY the standard harmonic Cmin AND Melodic Cmin scales.

WHAT IF the fact that in modern western music when they say 'Harmonic' vs. 'Melodic' minor scales are actually just missing notes of the one scale -
1  0  1  1  |  0  1  0  0
C  D  E  F  G  |  Ab Bb B  C

Which is relative minor of Eb - Eb / Cmin Traditional Harmonic Scale would be the same as this without the B natural.

Note that here when we break the 2 chords through the scale up as we did before, the chords become:

The Diminished chords remain the same allowing the same movements.

Now the Twist

If we take what we've just done with 1 0 1 1 (C D Eb F G) and rather than inverting them in the same direction, we invert it to 0 1 0 0 , but then go in a negative direction from the root, we get :
0  1  0  0 | 1  0  1 1
Ab A  B  C | D  Eb F G

Now take that negative section and tack it on to the original first section:

1  0  1  1 | 0  0  1  0
C D  Eb F  G | Ab A  B  C

We get the original Barry Harris Cm6 Dim scale - but built from contrary Motion.  When played in contrary, the notes 'just work' like this.  So Barry Harris' Cm6 Dim scale is the contrary version of the C6 Dim scale based on inverting the initial 4-bit sequence.

Taking it a step further, you can then take any 4 bit sequence and invert it, the either append, or run contrary to the root to derive many other workable scales other than just our standard major and minor ones.

Derived Chords from Contrary Min6 Diminished (Barry Harris)

Now we get these chords if we build them up in the same way as before:

Intervals and Chords Generated

As Barry Harris points out, through using these very simple 2 scales, now we can 'generate' any chord we need without even having to give the chord a name.  You use 'movements' to create sounds / emotions with the chords and through combining them over different bass notes, the permutations are numerous.

One of the remarkable things because of the symmetry, is how when played in contrary motion, regardless of where you start from - if you start on a root note (E.g. in C in a C6 Dim Scale), you could start on C, E, G or A, with your left hand and any one of those notes on your right hand and work contrary, the interval will be harmonically sound.  The only one a little 'harsh' would be the interval of note 5+6 together (G+A if in C).

C6 Dim Contrary Motion Intervals

Movements and Usage of Scales

So what scales can be used over what chords?

A lot of the time, it's not just an issue of seeing a chord and choosing a scale.  You need to see what role that chord is playing within the scope of the current Key.  E.g. - Dominant G (G7) if in the key of C will be resolving back to C, where if the key was G, you it would be playing the root role.  G's Diminished set is the 'Y' set (Eb, Gb, A, C), whereas C's is the 'X' set (D,F, Ab, B).  playing the latter over G would naturally resolve to C, where playing the 'Y' set over G would sound like it was going somewhere 'away' from G.

Min6 on 5th = Dominant 7th

Barry Harris likes to call the 6th on the 5th (Min6 on 5) when playing a Dominant 7th chord.  Ie. If you were playing C7, you could work around the Min6 Dim scale on G (which is the 5th of C) - on a C base, you'd have:

C || G Bb D E = C9 (note it includes the D and gets the 'rub')