Although they are often discussed, recorded, and published as a collection, the six concertos encompassing the so-called Brandenburg Concertos were not written all at once, nor for the same ensemble. Scholars suspect that Nos. 1, 3, and 6 may have been written much earlier than the others, perhaps dating from Bach's Weimar period (1708-1714), while 2, 4, and 5 most likely came from Cöthen. Bach later put the six concertos together and dedicated them to the margrave of Brandenburg, hoping to get a new job out of it. (He did not.) In fact, the only commonality among the six is the use of a three-movement, fast-slow-fast design; this indicates that the Brandenburg Concerti were based on Italian concerto format. Beyond that, they have nothing in common, and, in fact, among the six, there is as much variety as you can find in any six works by Bach.
Brandenburg Concerto No. 1 is, like all the Brandenburgs, set in five movements; the first three follow the typical fast-slow-fast arrangement of Italian concertos: here, allegro, adagio, allegro are indicated. The final two movements however, are additional dance movements, atypical of a concerto or concerto grosso of this time. The work is scored for two horns in F, three oboes, strings, continuo, and the violino piccolo. This was a smaller sized , 4-string instrument, similar to the violin, though tuned differently (The part is written in D Major, so it sounds a minor third higher than written). Bach uses the instrument here, in Brandenburg Concerto No. 1, in Cantata 140, and in a revision of Cantata 97 (Prinz). It is not entirely clear why Bach selected to use this unusual instrument here or in the other two works, since all the pitches Bach uses fall into the normal range of the "regular" violin. It could be that the timbre of the violino piccolo was distinct enough that Bach used it here to alter the "normal" violin sound.
Brandenburg Concerto No. 1 is a good example of a work inspired by the Italian instrumental composers Torelli, Albinoni, and Vivaldi. Typical especially of Vivaldi's concertos is what scholars and music students today call "Fortspinnungstypus" (Fortspinnung type). Here, an orchestral ritornello contains three distinct portions. "The first segment contains the tonality by focusing on the chords built upon the first and fifth degrees of the scale (tonic and dominant). The second segment follows with short bits of thematic material repeated at the different pitch levels called 'sequencing'); the changes in underlying harmony are marked mostly by successions of chords with fundamental pitches that are five positions apart in the scale. And the third segment brings the ritornello to a satisfying end by way of a closing gesture in the tonic." (Marissen in Oxford Companion to J.S. Bach)
You can see the similarities between the first and third excerpts - same key (F), focusing on the tonic (I) chord. In the third example, we haven't reached the dominant yet, but it's coming, I promise. In either case, the pitches F, A, and C play the prominent role, since these are the pitches of the tonic chord, and the pitches which most define the key. The middle excerpt is set instead in the relative minor key (vi), d minor. In this short excerpt, we don't get to see too much sequencing, but you can see some development and repetition of individual motives.
One other note in the excerpts above: if you return to the first excerpt, you can see two boxes marked in pink. These indicate the initial entrances of the two horn parts. though sometimes lost in the entire, busy texture of the strings and oboes, these passages have received a fair amount of attention from scholars, and may be a partial key to dating this work. They are, as Malcolm Boyd (among others) points out, hunting horn calls. Bach writes several pieces (including a few secular cantatas) around 1713 that use horn calls, and the presence of that same kind of writing here may help to date this piece to about the same time.
The second movement is a composing out of the first four measures, in which all of the thematic material for the movement is presented:
The fact that the first oboe and violino piccolo dominate melodically, the strings are relegated to a purely supportive role, and the horns are gone completely transforms the timbre and feel of this movement. What I find most interesting musically, however, is the very end. After a cadence in d minor (the tonic of this movement) four measures from the end, a series of dissonant harmonies leads to another cadence on the dominant (A Major). The third movement begins immediately back in F Major, and the tonal jolt can be unsettling.
The third movement, an allegro, is a courante style piece, set in 6/8. Here, the violino piccolo again plays a leading role independent of the orchestral violins; the horns, too, are prominent, adding strength to the texture, which may be why this movement has a feeling of being a finale. A brief, two-measure adagio interrupts the flow about two-thirds of the way through the movement; this is followed by a reprise of the opening material, and a strong return to F Major.
A minuet and trio follows to slow the flow a bit, and give a touch of grace to the concerto. A repeated horn figure heard throughout the third movement plays a prominent role in the minuet as well, providing coherence across those two movements. The trio is reeds only, providing a lighter texture and significant timbral variety to the concerto.
Brandenburg Concerto No. 1 closes with a polacca and trio. These, too, provide timbral and textural contrast. The polacca uses strings (excluding the violino piccolo) and continuo only. The trio (a different trio than was used in the minuet) is set for three "voices" or parts - two horns and unison oboes, meaning there is effectively no bassline in this movement.