For those whose brains hurt---sorry for the confusion--

Here is a bit of clarification for Frank's point--

When he talks about a chord being mislabelled and how it officially works--he really means finding a label for the chord that makes what is happening in terms of music theory--

He uses the chord C#° as an example--

To make it easier--here is the kind of chord progression that it would occur in: C/ C#°/Dm/G7--

So the question is--why does that C#° chord make sense in a chord progression that otherwise seems to be in the key of C?

Before we get to Frank's answer, remember that we've talked about the fact that the Dominant Seventh Chord(G7, A7, D7, etc) contains a diminished triad--

G7 is G-b-d-f   and the b-d-f are a diminished triad--

So if you take a diminished triad, such as b-d-f-, and add a note that is a major third (two major scale steps) below the root of the diminished chord (which, in this case, is b) you get G-b-d-f--which is G7--

Frank points out that generally, in chord notation, the diminished chord is usually a diminished 7th--

Frank's example is a C#°, which would be C#-E-G-Bb--so when we add the major third below the root of the diminished chord, we have A- C#- E -G- Bb, which is an A7b9 chord--

Now if we substitute this chord, we end up with this progression: C /A7(b9)/ Dm /G7--which is a circle of fouths chord progression that we all understand--

(I put the b9 in parens because, as is always the case, those extra notes are flavoring, and can be left off without altering the chord progression)

This means that when you see a diminished chord, such as B° or C#° in a chord chart, you can simply turn it into a dominant seventh chord by adding the note two major scale steps below, and play the G7 or A7 instead--