Just checking my understanding. A diminished 7th is made up of the tonic, minor 3rd, diminshed 5th and diminsed 7th - is that correct?

So a diminished 7th starting on D would be:
D natural
F natural
A flat
C flat

I'm only confused because in my scale book this has been written as D F G# B. These are the enharmonic equivalents of the above notes, but obviously have different names as an interval. I just wanted to understand where the notes were coming from...

Yes, as far as I know a diminished 7th is made up of all minor thirds. Sometimes in scale books they are written enharmonically because the notes are more familiar.

Spelled that way it's really and inversion on a dim7 on G#.

Thank you - glad to know that it's their spelling and not my understanding at fault here

A diminished seventh chord can work as a functional or embellishing harmony. As a functional harmony, the root of the chord works as the leading note.

The chord D-F-Ab-Cb is therefore the VII7 in the key of Eb.

The chord D-F-G#-B is therefore the second inversion of the chord G#-B-D-F, which is the VII7 in the key of A.

Seconded. Couldn't have explained it better.

However, I must ask: Did your book explicitly say "D F G# B is an inverted diminished 7th" or did it leave you hanging?

The books is the AB Scales and Arpeggios book, so there are very few words in it - just the title of each scale/arp.
So it's entitled 'Diminished 7th starting on D two octaves'. That's all the writing it comes with.

Interestingly (perhaps), this was in the descant recorder part of the book; I haven't looked at the dim 7ths in the treble recorder part of the book to see if they are written differently. There is only one dim 7th for descant, so it's not like there is a lot to do comparisons on though...

Ut-oh. Sounds like a typo.

I don't play the recorder, but I wonder if you would approach the pitches Ab and G# differently, even though the arpeggio would be played out of context (i.e. without having it resolved to a consonant chord). Could this be the reason for spelling the chord with G# and not Ab?

I think that they maybe just spell it in what seems to be the least complicated way rather than the strictly accurate way.
Just looking in the grade 8 piano scales book and the same thing happens there for a lot of the dim7 arpeggios. They are not all spelled correctly as dim7 chords in their own right but as inversions in most cases.

Er, how?

Probably because it involves the fewest accidentals! And in acknowledgement of the fact that those taking G5 descant recorder are quite likely doing so in short in-school lessons where you don't have time for the theoretical discussions.

It's great for me as I find sharps easier to deal with than flats, in general, so seeing G# gets me the note a bit quicker than Ab. Actually, I'd say that one's quite even, but F# I could do waaaay quicker than Gb. Silly, I know, but just the way my mind works (Bb is the one 'definite flat' note that I have - I'll find Bb quicker than A#. Eb and D# are about even, as are Ab/G#. F# and C# are more 'definite sharp', I'll find them quicker than Gb or Db).

I very rarely come across C-flat, and so having it written as B is just easier. Likewise Fb, E# or B#. Just don't see them often enough for my mind to compute it

An 18th C flautist could well have different fingerings for the two notes (possibly depending on which flute method he used; I don't know that all their fingering charts did this). I don't usually change recorder fingerings with the notation, but I might well do so because of the context: extra fingers either because I'm playing the third of a major chord or because my part should be louder.

I might need some help here. I was teaching the dim 7th from Bb to my adult TG Grade 6 pupil yesterday.
In the TG Scale Book it is written out as Bb Db Fb Abb. She also has the AB Scales and Arpeggios which has it notated as Bb C# E G which is of course much easier to read.

The second is the third inversion of vii7 in D minor. What label would you give the first one with the Abb? Which would be the correct or best one to use when teaching and explaining?

Erm.... vii7 in Cb?

It would depend on the level of theory knowledge needed or that can be understood which to use I think.
If it's just for the purposes of learning how to play a dim7 arpeggio on that starting note then I don't think it matters too much. If it's piano then just knowing that it's a series of minor 3rds would make it ok to work out without the notation anyway.

Like I wrote earlier, think of the root as the leading tone. Bb is the leading tone of Cb; therefore, the chord Bb-Db-Fb-Abb is the viio7 in the key of Cb.

Thank you organ_dummy. That makes perfect sense. More useful to me doing advanced theory than for my pupil. As sbhoa says I always teach dim 7th by sound and stacking minor thirds, but my pupil has both books and I like to give clear explanations - of why there are two different notations.

You are vey good at chord analysis - I can do it but I just need more confidence that I have the right answers sometimes Do you do a lot of composition?

Thank you for your compliment. I'd like to return to composing some day; right now I simply don't have the spare time. Teaching theory classes is my day job.