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Why did Beethoven write a leading-tone IAC here? It appears as viio 6/5 - I. Also, why does the leading tone IAC exist? enter image description here

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    Maybe these should be two different questions. I do think both are good questions, though!
    – user45266
    Feb 12 '19 at 17:32
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    Leading tone 7th chords are very common, especially from Beethoven onwards. Are you talking about from measure 140 to 141? Because I see a diminished 7th going to tonic there. It is probably just there as part of the picardy third setup to go from C minor that the piece has been in to an ending in C major(as I can tell by all the E naturals). The diminished 7th probably in this case gives smoother voice leading from C minor to C major than a dominant chord would. Or maybe it is just for the purpose of a dramatic ending. I know that Beethoven uses the diminished 7th a lot for drama in his works
    – Caters
    Feb 12 '19 at 17:39
  • So the V chord is replaced by the vii chord! Good point! That Bdim7 is like a G7b9 without its root!
    – user53472
    Feb 15 '19 at 1:36
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This is a coda at the end of a movement. The (sort-of) PAC moment that concluded the recapitulation occurred right before the excerpt given here.

In codas, it is very common (even sometimes before Beethoven) to have a series of moves to the subdominant and/or dominant, often all over a tonic pedal (or something like that), without another strong root-position PAC. Beethoven frequently makes use of V or viio chords over a tonic pedal at the of movements in coda or coda-like passages. See, for example, the end of the Moonlight Sonata, which alternates harmony between V and i. Or, for an example with a Picardy third (as here), see the end of the first movement of Op. 49, No. 1, which repeatedly moves viio7-I in the closing bars over a tonic pedal.

While the tonic note isn't always sounded on the beat in the excerpt given in the question, it's pretty pervasively emphasized in the texture and functions as a pedal note throughout this ending.

As noted in comments, the viio is just a dominant function chord, and it tends to be frequently used in such coda passages over a tonic pedal as the upper voices often tend to move in neighbor-tone motions. Sustaining the fifth above the bass (the root of V) isn't as interesting and doesn't generally follow the pattern of the other voices, so having it move up to the flat sixth above the bass (effectively creating a viio7) is rather common in alternating with the tonic harmony, particularly in minor key pieces.

(This perhaps gets at an answer to the OP's question of why a "leading tone IAC" exists -- it's not a strong cadence in classical style, but tends to have strong voice-leading motion with all voices moving in strong stepwise motion toward the tonic chord.)

I'd also note the pervasive use of diminished sevenths in the harmonic vocabulary within this first movement of Op. 111 in general. Diminished sevenths are an integral part of the opening and of some of the primary thematic material, so it's not surprising that Beethoven would choose to bring out that sound at the end of the piece as well.

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