I would call this a Baritone-Tenor.
Four whole steps lower than a t/t.
I would not.
It's what the Wheatstone ledgers call a baritone-treble.
four whole steps lower than a tenor-treble, but that's not the same as every note being pitched exactly that amount below the equivalently located note on a tenor-treble.
On the picture of the left hand side you can see no more than 4 extra buttons below mid-TS.
- On a Treble middle C is on the left hand side in line with the middle of the thumb strap. (Call this position "mid-TS")
- On a Tenor mid-TS is F below middle C.
- On a Baritone mid-TS is C below middle C.
- On a Tenor-Treble mid-TS is middle C, like a Treble, extended below mid-TS with 4 buttons on each side, going down to C below middle C on the right hand side.
- On a Baritone-Treble mid-TS is a middle C, like a Treble, extended below mid-TS with 8 buttons on each side, going down to F below C below middle C on the right hand side.
If the bottom note is F below C below middle C, than mid-TS is F below middle C, like a Tenor.
No, the naming convention has nothing to do with what note is next to the thumbstrap, except incidentally. (Would my baritone-treble suddenly become a tenor-treble if I moved the thumbstraps "up" a row? I don't think so.)
It has to do with where the notes are placed in terms of which end of the instrument and which side of the center line of the button array. If in those terms the notes are placed the same as on a standard treble, then the instrument is either a "treble" or a "[something]-treble". The something then indicates how low the instrument's range extends. A "tenor-treble" extends downward from the treble range to C-below-middle-C (the low note on a viola, or "tenor" viol). A "baritone-treble" extends that pattern down even further. For the same fingering, a standard "baritone" sounds an octave lower than a treble, and a "bass-baritone" extends the baritone
pattern down to the bass range, so the whole thing is an octave lower than the tenor-treble.
The importance of all this is not in the name:
When you're used to play a Treble, a Tenor-Treble or Baritone-Treble gives you just the extra notes at the lower range.
When you play the same music on a (Baritone-)Tenor, you have to transpose.
Well, I've never heard the term "baritione-tenor" before your post, Leonard, but I assure you that on my own "baritone-treble", 1) the note inboard of the center of the left-hand thumb strap is F below middle C, and
2) it is most definitely not
a transposing instrument.
On my 64-button baritone-treble the buttons
are in exactly the same placement relative to the thumb straps as on my 64-button tenor-treble, but the notes
are not. And I repeat, it is not
Your theory about the relationship between the location of middle C relative to the thumb strap and transposing instruments is interesting, but I have yet to encounter such a transposing instrument, yet I have handled more instruments than my own which provide counter examples. Have you really seen what you call "baritone-tenor" concertinas, i.e., with buttons laid out like my baritone-treble, but with the central columns of buttons not all notes of the C scale? And how many transposing "tenors", for that matter? An old friend of mine had a 48-button "tenor" (I think Wheatstone would still have called it a "tenor-treble"), and it was not transposing.