I had considerably diminished the use of the diminished scale in my own playing for quite a while now, since I've felt that the modes of Melodic Minor (which is 2 notes shy of being a diminished scale), primarily the "altered scale" (seventh mode, diminished / whole tone) is a more interesting and flexible scale choice for most of the mutually applicable harmonic situations.
Plus, the diminished scale being a synthetic, symmetrical scale, can sound so...................diminished, synthetic & symmetrical.
Then there is that handful of cliche licks, probably less than ten of them, that have been "regurgitated ad nauseam" (or is it the other way around?) since their introduction into the jazz vocabulary sometime in the mid 1950's (I think) when they were actually hip.
I know, too, that I'm one of the guilty parties. I guess the symmetry of it made it convenient, if not exactly "easy", to learn and play. It was one of the first "outside sounding" devices I learned back in the day and I think I beat it to death eventually, at least for my own taste.
But now.....I think it's time to confront my own prejudices!
Back to the perfect 4th thing. The diminished scale contains 4 sets of perfect 4ths, spaced a minor 3rd apart. Using the C half tone / whole tone diminished scale (C-C#-Eb-E-F#-G-A-Bb), the first in the PDF example, this means:
C# - F#, E - A, G - C, Bb - Eb
Instead of playing them a minor 3rd apart, let's put them a tritone apart.
C# - F#, G - C & E - A, Bb - Eb
creating two 4 note groups a minor 3rd apart.
Each measure of 8 eighth notes in the exercise covers all 8 tones of the diminished scale. Each line contains a different juxtaposition, or melodic shape.
Each 4 note grouping could also be seen as two tritones a P4th (or P5th) apart,
C# - G, F# - C
in our example. Some harmonic possibilities for this configuration could be:
A7 #9 13, Eb7 #9 13 (from the P4th configuration) as well as D 11 & Ab 11 w/ both Maj. and min 7ths (from the P5th configuration).
This is a suspenseful sounding group of notes, both melodically and harmonicalli and will actually work over any bass note, depending on your taste and tolerance for dissonance.