A diminished scale is an eight-note scale, or octatonic scale, made up of a series of alternating whole steps and half steps from the starting pitch. One way to make sense of these scales is to use a diminished 7th arpeggio as their foundation.
The Eo7 arpeggio below has the notes E, G, Bb & Db. Each chord tone is separated by a minor 3rd (3 half steps).
‘Filling in’ each minor 3rd of the arpeggio with an additional note creates an alternating pattern of whole/half steps or half/whole steps, depending on the placement of the additional notes.
Adding a note a half step above each chord tone creates a diminished scale with a half/whole step pattern.
Adding a note a whole step above each chord tone creates a diminished scale with a whole/half step pattern.
Focusing on the chord tones of the foundational arpeggio, in this case E, G, Bb, Db of E07, makes it easier to visualize a diminished scale along the fretboard. The exercises below use the notes of both E diminished scales to embellish an Eo7 arpeggio.
Ken Pasciak 
The idea of adding a note a half step above each chord tone to create the half-whole scale is a neat mental shortcut—way easier than memorizing eight separate notes. Always struggled to visualize diminished scales on the fretboard, but breaking it down through the Eo7 arpeggio really clarifies the pattern.
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