The half-whole diminished scale is just the regular diminished starting on the second degree. In jazz, it’s one of the two most widely used scales over a dominant 7th chord, the other being the seventh mode of the melodic minor scale. Some of this can also be infused into blues playing - these scale substitutions are among the ways of jazzing up the 7th chord and extending your palette of blues hues.

Let’s lay out an Ab diminished scale in quasi-thirds, taken from the viewpoint of a G7 chord:

G B D F Ab A# C# E G# B
1 M3 P5 m7 m9 +9 +11 M13 +15

Now this may look a little strange, what with the upper G# being the same as the lower Ab, but hold your fire for a minute.

The first four notes give us plain old G7. Everything else is extensions to this base chord. We can add a lowered 9th (Ab) or a raised 9th (A#). The latter sounds like a minor third and gives us that simultaneous major/minor feel. It’s a dark, tough-sounding chord when played aggressively, but can also sound very mysterious when used contemplatively (as in Miles Davis’ “In a Silent Way”.

We’ve also got a raised 11th (C#), a major 13th (E), and even a raised 15th (double octave) in G#, if we want to look at it that way. Why a raised 11th instead of a flat 5th? Because you can
have a regular fifth and a raised 11th at the same time, and the effect can be suspended and luminous if desired, instead of the feeling of grinding the strong imperious fifth down into a
flattened and funky grovel.

Note that I’ve spelled out an E major triad at the top of the stack. This can be arpeggiated or played as a chord above the G7, for an other-worldly sound. Check out the intro to Miles Davis’
recording of “Someday My Price Will Come.” The pianist is playing F7 in the left hand, and arpeggiating D major in the right - the same thing. Now, you could just look at this as a chord
superimposition, but if you want to fit it into the background of a parent scale instead of just treating it as an inert phenomenon, the diminished scale is the logical place.

One of the interesting things about a diminished scale is that it is symmetrical - it is based on a smaller interval pattern that repeats without variation to complete the scale. In the case of
the diminished scale, it’s the pattern of Tone-Semitone, which repeats at intervals of 3 semintones (which may be a minor third or an augmented second, depending on the spelling);

Ab Bb B C# D E F G Ab
T S T S T S T S

What this means is that anything you construct from a diminished scale can be repeated exactly at three other places in the scale. For instance, over that G7, we can play a G major triad, a Bb, a Db, and an E (like we just looked at). Each one of these superimpositions gives a different flavor and a different degree of tension.

One way to look at a diminished scale is to break it up into two diminished 7th chords that, depending on where you start, are either a semitone apart or a whole tone apart:

Ab B D F
G Bb Db E

Again, from the viewpoint of G7, we can play the Go7 (diminished 7th), whcih puts everything but the G a seimtone too low, then slide it up a semitone to Abo7, putting everything else in tune, but
raising the G a semitone, giving a lowered 9th. This gives an incomplete resolution that can be used to create a perpetual state of varying tension.

Of course we can also play the regular G7 and the Go7 simultaneously by extending the chord out in thirds, as done above. Try this on a piano and you’ll get a sound familiar from TV scores and big band arrangements.

Let’s go back to that raised 9th chord:

G B F A# (the 5th is omitted)

and take the last 4 notes up a semitone, and put them over a D bass:

D C F# B

We now have a D7th chord with an added 13th, a very sunny sound. And yet this sunny little chord sounds perfect with the diminished scale, and much more interesting than if we played it
with a D mixolydian mode or a D pentatonic. Of course, if we put an Ab under the top three notes, we get an Ab raised 9th chord:

Ab C Gb B

The two roots, D and Ab, are a tritone (diminished fifth apart). (these are just two of the four possible roots under the symmetrical-repeat viewpoint, the others being F and B). So we can play the same chord in the right hand (or on guitar) while varying the bass note between D and Ab.

We can do the same to the G +9 chord, varying the bass note between G and Db, again getting the sunny chord over Db instead of the tough chord over G.

G B F A#
|
Db (Cb F Bb - same notes, spelled for Db root)

And the Db note can be a semitone lead-in to D, while the Ab can be a lead-in to G. Let’s look at playing just two chords (the G and the D) in the right hand while looping the bass part around
with this tritone thing:

G B F A#
| | | |
Db | | |
| | | |
D C F#B
| | | |
Ab | | |
| | | |
G B F A
| | | |
etc.

This can be extended chromatically all the way around the cycle of 5ths if you want to break out of this one scale.

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