Am, F, G, Em

[jancivil]

Can you clear up some misapprehensions I may have had about chord names here...

A diminished seventh here will be B double flat, almost always off a diminished triad, eg., C Eb Gb Bbb. (*diminished, diminished seventh*)

when you say Bbb, is that equivalent in pitch to an A? If not, what does it mean? If so, I have always just called this chord a 'diminished', and thought of it as having minor 3rd, flat 5th, and a 6th rather than a double flattened 7th. Is there a reason I shouldn't think of that note as a 6th?

It's a tertial harmony, the convention is tertial harmonies. All of the intervals are minor thirds. It's a diminished seventh chord. There are reasons for calling it that.

You want to call Bbb 'A'. Ok; that one spelled from A is A C Eb Gb. Let's take that spelling.
The reasoning also has to do with key. This particular construction, the diminished seventh chord, happens in minor with a raised seventh as a dominant* type of harmony, known as vii7.
IE: Scale of harmonic minor: Bb C Db Eb F Gb A.
The Aº7 is vii7 according to that: A C Eb Gb.

Dim7 is symmetrical, there are four facets; so for chromatic harmony it's useful to see all four versions/spellings, and their particular dominant proclivity.

If you're consistent with calling Bbb 'A', you want to call that Gb 'F#'. Let's do the same I did above: F# A C Eb. It's not an F# in that Bb minor event because Gb is naturally occurring in key; F# doesn't belong.
F#º7 works as vii7 in G minor, off the raised seventh. Call that dim. seventh Eb a sixth, D#, that D# doesn't belong. If you call it D# you're connoting vii of E minor. D# F# A C.

* It's quite like V7 with a flat 9. Note the tertial structure. In Db: Ab C Eb Gb Bbb; in Bb: F A C Eb Gb; in G: D F# A C Eb; in E: B D# F# A C; in C#: G# B# D# F# A.

Now you can look at it as a modulatory device through the new spellings. The practice is behind the reasoning of the spelling.

And, I always called the chord you mention below a 'diminished seventh', as it had a 7th instead of the 6th I talk of above. I have heard of half diminished but thought it synonymous with diminished seventh.

'Half diminished', eg., C Eb Gb Bb, is a *diminished, minor seventh* chord. Aka minor seven flat five chord.

No, half diminished is 'half-' vis a vis full diminished, it's diminished triad/minor seventh. m7b5. ii7 in minor.

[someone called simon]

It's a tertial harmony....

Took a little getting my head around that, but I see the reasoning. Thanks for the explanation.

[jancivil]

Break the chord: A C Eb Gb, make it single notes and horizontal. Minor thirds, it's simple. Eb to F# is an augmented second. Or, the augmented second inverted, F# to Eb is a DIMINISHED SEVENTH.
Tertial = thirds. The whole thing it comes out of is all chords made of thirds, this can't be any different, that's the convention.

So, a composer giving such a spelling as A C Eb F#, given that this is the position or voicing, could be indicating the ambiguity or possibilities of this harmony (ie., this is first inversion of F#º7). It may be she is simply going for convenience and the assumption is everyone knows it is what it is. But in terms of teaching the diminished seventh, that interval is its own thing rather than be reduced to 'same as major sixth' conceptually.