Elaborate on the formal structure of sonata form, including the exposition, development, and recapitulation, and detail the typical harmonic plan and thematic treatment within each section.

Sonata form, a cornerstone of classical music, is a formal structure that provides a framework for musical expression, particularly in instrumental works. It is typically comprised of three main sections: the exposition, the development, and the recapitulation, each with its own distinct characteristics in terms of harmonic plan and thematic treatment.

The Exposition: The exposition is the first major section of sonata form. Its primary function is to present the main thematic ideas of the movement and establish the tonal relationships that will be explored throughout the piece. The exposition typically consists of two main thematic groups, each with its own distinctive character, and these themes are presented in contrasting keys.

The First Theme Group: The exposition usually begins with the first theme group, which is in the tonic key, the home key of the piece. This theme is often a clear, assertive, and memorable melody, and is also known as the primary theme. It can be a single melody, or a collection of related melodic ideas. The tonal and harmonic nature of the first theme is usually stable and firmly rooted in the tonic. The first theme group establishes the fundamental character and overall mood of the movement.

The Transition (Bridge): Following the first theme group, a transitional passage, or bridge, is presented. The transition serves to modulate from the tonic key to a new key, typically the dominant (in major key pieces) or the relative major (in minor key pieces), where the second theme group will be presented. The transition section often involves thematic development and use of unstable harmonies that creates tension and a sense of anticipation. It often increases in intensity, preparing the way for the more lyrical nature of the second theme.

The Second Theme Group: The second theme group is typically presented in the new key (dominant or relative major). The second theme is usually more lyrical and contrasting to the first theme. It is also known as the secondary theme. The second theme is designed to provide a sense of contrast and is often more melodic or song-like than the primary theme. It might have a softer dynamic, or slower tempo. The second theme group can also include several related melodic ideas that are built upon the original theme.

The Closing Theme (Codetta): The exposition often concludes with a closing theme, or codetta. This section is also in the secondary key (dominant or relative major) and reinforces the newly established tonal center of the second theme group. It provides a sense of closure to the exposition and prepares the way for the development section. This section does not typically include any new thematic material. It's more like a summarizing section, and it's often shorter than the first and second theme groups.

The Development: The development section follows the exposition and is characterized by the exploration and transformation of the thematic material introduced in the exposition. In this section, the themes are fragmented, combined, transposed, and otherwise modified.

Thematic Development: The development focuses on working with the themes, motives, and melodic fragments presented in the exposition. Themes and motives are often passed between different voices, or altered rhythmically, melodically or harmonically. This section is a musical playground where the composer uses elements from the exposition to create new ideas and musical textures. It often modulates through many different keys, creating a sense of harmonic instability, and it is a place where the tension builds up, setting the stage for the recapitulation.

Harmonic Instability: The development is characterized by a lack of harmonic stability. The themes are often presented in different keys and using dissonant or unstable chords. The purpose of this is to move far away from the tonic key and to create a sense of tension. The development can often sound fragmented and incomplete, and the harmonic instability adds to this. The development also has no clear structure and there is no typical form here.

Transition Back to the Tonic: Near the end of the development, the music often moves towards a dominant pedal, where the dominant chord of the main key of the composition is held for a prolonged time, creating tension which resolves into the tonic key for the recapitulation. This prepares for the return of the tonic key and signals to the listener that the development is over and a new section is about to start.

The Recapitulation: The recapitulation is the final section of the sonata form and is essentially a restatement of the exposition. This time, all of the thematic material is presented in the tonic key, creating a sense of closure and resolution.

First Theme Group in the Tonic: The recapitulation begins with the first theme group, this time presented in the tonic key. The material is generally the same as in the exposition. The music is more stable and now there is no tension as before in the exposition.

Transition (Modified): The transition section is also repeated, but modified this time. It does not modulate away from the tonic like it did in the exposition, but stays in the tonic key. The main purpose of this is to remain in the tonic key and prepare for the presentation of the second theme in the tonic.

Second Theme Group in the Tonic: The second theme group is now also presented in the tonic key. This is the crucial difference from the exposition, where the second theme is in the secondary key, it is now in the tonic key. The use of the tonic for both the first and second theme groups provides a strong sense of resolution.

Coda: The recapitulation is often followed by a coda, which is an extended closing section that confirms the tonic key. It reinforces the sense of closure and brings the movement to a definitive end. The coda is where final thoughts are given before the movement comes to an end, and provides the opportunity to finish the piece with the desired intensity.

The harmonic plan is therefore crucial for the form of the sonata. The journey away from the tonic in the exposition and development and finally back to the tonic in the recapitulation creates the dramatic arc of the piece. The tonal plan is not a rigid rule however, but is a general framework. It provides composers with a powerful way to create musical narrative. The tension and resolution created through the modulation, thematic development, and recapitulation creates a satisfying and complete musical journey.