I was listening "Brandenburg Concerto N° 4" and "Concerto for 2 Harpsichords in C minor, BWV 1062", and wondered: aside from the catalog number (that came later), did Bach name his pieces? Did he specifically write "Concerto for 2 Harpsichords in C minor" ? Or, perhaps those names were chosen by the people who created the catalog based on the instruments available in the works ?

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Well, some names are certainly contributed by editors and tradition (he never called his two-violin concerto a "double concerto," let alone, for obvious reasons, "The Bach Double"). But looking at his most famous violin works, the solo sonatas and the partitas, yes, he provided a title page to the whole work (Six solos for violin without bass accompaniment, Book 1), and for the individual pieces ("Sonata 1 for solo violin without bass").

Part of the point here is that the notion of a title as a part of the overall work evolved in a later period. For Debussy to name his tone poem "The Sunken Cathedral" indubitably contributes to our experience of the piece. But in the Baroque, the art object was the performance (and it was less of an Object at that). Bach churned out oratorio scores weekly, and referred to them simply by their first line or by the Sunday they were intended for. The Brandenburg concertos were a gift and so had an elaborate presentational title page, but that page was directed to an audience of one, not to the general public, and any title served merely to explain the contents. For the purposes of Bach, his copyists, and his musicians, all that was needed was enough of a signifier to tell one piece from another ("Concerto. No, for harpsichord. No, in A minor. No, the other one.").

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    It should be noted that Debussy in fact placed titles like "The sunken cathedral" after the score, specifically to avoid them influencing the interpretation of the pianist. Sep 1, 2021 at 7:31

Bach did give names to (most of) his pieces, but these are not necessarily the names used at present. The now so-called Brandenburg concertos are identified on the title page of the autograph as “Six Concerts à plusieurs instruments”, and no. 4 specifically as “Concerto 4to à Violino Principale, due Fiauti d'Echo, due Violini, una Viola è Violone in Ripieno, Violoncello è Continuo”.

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