: Diminished Seventh Arpeggio? Checking out something in the ABRSM manual of scales and arpeggios, I note that those 'formed of the chords of the dominant seventh' start and finish on the dominant note (In C, goes G-G). That

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I think the important distinction here is that between "the diminished-seventh chord on C'' and "the diminished-seventh chord in the key of C."

You're correct: the diminished-seventh chord on C has B??. But the diminished-seventh chord in the key of C is going to be b°7: B D F A?. This is because, when one considers the diatonic seventh chord, the only fully diminished seventh chord is that built on the leading tone.

But note that this suggests either C minor or the joint C major/minor (that is, modal mixture is included), because the chord build on the leading tone in a major key is actually half-diminished: B D F A, or bø7.

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The dominant 7th-shape chord that can be built using the notes of a C major scale is G7. There is no diminished 7th chord that can be made exclusively from those notes, but the diminished TRIAD that can is Bdim. So I guess that's why they lumped Bdim7 into the 'C major' group.

It's arguable that the most commonly-used dim7 chord in a piece in C major is C#dim7, in the ubiquitous C, C#dim7, Dm7, G7 sequence.