The diminished scale

Major's steps come in an uneven order that points home. The diminished scale strictly alternates: W H, over and over. Compare the two, then hear it over its own chord.

Because the pattern repeats every minor third, starting the scale on C or on E♭ gives the same shape, and nearly the same notes. Like the °7 chord inside it, it has no single root: four starting notes, one scale.

Count them: eight notes, one more than any key owns. The extra note is the giveaway that this scale is built on symmetry, not on a tonic.