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I am composing a piece that I have nicknamed "The March of a Hero" and I noticed something. If I separate the piece out into sections, I get this:

Introduction - A few slow arpeggios to introduce the hero

A - March theme, comes back multiple times, each time a bit different, Key is firmly in D major - The hero is accumulating his army

B - Lyrical and melodic, even the bass is melodic, giving rise to 2 voice counterpoint in what has otherwise had a homophonic texture, Key has moved to C major, close only by proximity to the original D major - Army has stopped to look and see what direction the enemy is approaching from

A' - March theme again, but richer, again with variation - The hero is accumulating a big army now

Bridge - Slow theme in C minor, now things are getting dramatic - The enemy's approach

C - Acts kind of like a development section, but with new material, Rocking bass is now representing the charge, Short motives are representing the hoofbeats of the cavalry, Fast block chords are representing cannon shots, Full blown battle theme, Key is in C minor

B' - A Picardy third turns things around in the hero's favor, we are now back in C major with the 2 voice counterpoint

A'' - March theme, but not in the expected D major yet, starts in C major, and then moves to C#major - The hero is coming home

A''' - March theme for a final time in D major, a victory fanfare, -The hero is home and he has saved the day

Coda- Final few chords to represent the hero's gratitude

As you can probably tell, this acts very much like a Rondo. But, it also kind of acts like a Theme and Variations, because in all except the final A section, there are multiple variations upon that March theme.

So, is it possible for a Rondo to also be a Theme and Variations? Or are those 2 forms not possible to have simultaneously?

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A "theme and variations" is typically just that -- a theme, and then variations on that theme. If you have multiple other sections (B, C, bridge) that have no clear relationship as a variation on that theme, then it wouldn't typically be a theme and variations.

That's not to say that modern "theme and variations" don't take more liberties with the form. And even historical theme and variations movements might sometimes incorporate a sort of "bridge" section between variations or a coda, though any bridge section would typically function mainly as transition, not as a full-blown section with its own theme.

A rondo, historically, tended to have multiple returns to the primary theme ("A section") in the tonic key. (Note the latter qualification: one thing that differentiated the classical rondo from the baroque ritornello form was the fact that the A theme always comes back in the tonic key.) Classical rondos also had all sorts of other standard conventions that aren't necessarily just about the returning A section.

In any case, it's your piece, so you can call it what you want. Your description doesn't sound much like a classical rondo, but that form was greatly expanded and modified by romantic and later composers to the point that the structure is really loose. It might be more of a stretch to call a piece a "theme and variations" when it has multiple contrasting sections that don't contain the primary thematic material. Also, calling it a theme and variations implies that the other contrasting sections aren't really important to the form (merely bridges/transitions, etc.), but your description seems to indicate that you think the B, C, etc. sections are essential to the narrative structure of the piece.

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While I wouldn't call it a Rondo That Is Also A Theme And Variations at this point, you can easily write a rondo where every return of the A section is varied. The 4th movement of Chopin's Piano Sonata No. 3 in B Minor is one such example (heck, the first return of its A section is in E minor instead of B minor, and the movement has aspects of sonata-rondo form). Liszt's "Mazeppa" D minor etude in his Transcendental Etudes is another example.

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