The String Quartet No. 15 in A minor, Op. 132, by Ludwig van Beethoven, was written in 1825, given its public premiere on November 6 of that year by the Schuppanzigh Quartet and was dedicated to Count Nikolai Galitzin, as were Opp. 127 and 130. The number traditionally assigned to it is based on the order of its publication; it is actually the thirteenth quartet in order of composition. Music The five movements of the quartet are: The performance of the work takes around 45 minutes. I. Assai sostenuto – Allegro The slow introduction to the first movement, like that of the thirteenth quartet, is based on a motif that recurs throughout the late quartets and in the Große Fuge as well: the second tetrachord of the harmonic minor scale. The movement is in a modified sonata form which involves three full rotations of the expositional primary and secondary thematic material, each with a different tonal plan, in contrast to the usual sonata form which only cycles fully through this material twice (for the exposition and recapitulation). The first expositional rotation begins in the tonic and moves down a third to VI (F major) for the second key area (m. 48), while the second – following a developmental episode – is almost a direct transposition that begins in E minor (m. 103) and moves similarly down a third to C major. The final abbreviated rotation (m. 193) remains in the tonic throughout, followed by a coda (m. 232) featuring a dominant pedal point. The movement's unusual structure has invited a variety of analytical interpretations. Composer Roger Sessions describes the form as more of a triple exposition than a normal sonata form, and the second rotation could be interpreted as a simulation of the expositional repeat seen in many classical sonata form expositions, with the added interest of transposition. Conversely, other analysts have interpreted the second rotation as the onset of the recapitulation or as a "double recapitulation effect" rather than as an expositional repeat, with Hepokoski and Darcy describing it as a "tonally 'wrong' recapitulatory rotation followed by a notably varied, 'right' one in the tonic". and Joseph Kerman referring, albeit with reservations, to an "E-minor recapitulation" and an "A-minor recapitulation". Charles Rosen, on the other hand, considered this structure to be governed, as all of Beethoven's works, by the principle of sonata resolution, pointing to Haydn's 75th and 89th