In the context of musical form, the term binary means a formal type that has two main parts often called reprises because each main part is typically repeated. There are two types of binary form: rounded and simple. Both forms have the possibility of featuring a balanced aspect as well (note: balanced binary is often described as its own type of binary form but that approach is not taken here). Binary forms are common in European classical-style music from the 17th, 18th, and 19th centuries and they were used heavily in dance music in that style. Binary form is typically one of the shorter forms and because of that, they are often embedded within larger, compound forms like compound ternary form.
In 17th- and 18th-century classical music, each reprise of the binary form is typically repeated in the following fashion:
Each reprise is immediately repeated so that the listener will hear the following layout of parts: Reprise 1, Reprise 1, Reprise 2, Reprise 2. In 17th- and 18th-century music, it is very common to find the repeat signs written in the score and decorative improvisation was expected when playing the repeat but was not specifically indicated in the score. In the 19th century, it became even more common for composers to write out the repeat explicitly in order to indicate specific decorations and/or to include changes in some musical domain like instrumentation, register, or possibly to expand the music beyond the length of its first statement. It also become increasingly common in the 19th century to remove the repeat of one or both reprises.
While having two—usually repeated—reprises is common to all binary forms, there are two relatively distinct subtypes that capture the form's larger melodic organization. Those subtypes are rounded and simple, and their first section is represented with the letter A (the work's main section). Formal organization is represented with upper-case letters and sometimes prime symbols (see Adding Letters and/or Generic Labels for more details). The A section contains the form's main musical material and the beginning of A contains the material you would likely sing to someone if they asked you how a piece went. B sections vary depending on the type of binary form (more details below). Both forms can also feature a balanced aspect (represented with an x in parentheses in the diagram below). The basic overview of these subtypes is represented below:
In rounded binary, the beginning of A returns—in the home key—somewhere in the middle of the second reprise. It is not necessary for all of A to return (though often it does), only the beginning. While the returning material may be exactly the same, it's also common to see slight variations, like change of octave, accompanimental pattern, and/or melodic embellishments. If there is variation, you should still be able experience the feeling of return when the A material comes back. If unsure, you can expect the harmonic analysis to remain essentially the same, the chord changes will likely be in the same metric locations, and the scale degrees of the melody will also be in the same order and in the same metric locations, just make sure to account for the possibility of slight variation in the domains listed above. In rounded binary form, the second reprise starts with a B section. Typically, the B section is less stable than the A sections and may involve common destabilizing features like sequences, chromaticism, and dominant pedals. In some binary forms, however, the B section is quite stable but simply presents different thematic material than A (see, for example, the B section from the Trio from the third movement of Mozart's String Quartet in G major, K. 80).
The menuetto of the third movement of Mozart's symphony no. 25 in G minor (Example 1), is a clear instance of a rounded binary form typical of the middle-to-later 18th century. After a relatively stable thematic statement (A) during the first reprise (mm. 1-12), the second reprise (mm. 13-36) can easily be divided into two distinct parts, B (mm. 13-20) and A′ (mm. 21-36). The impression of a division is the result of the return of A material at m. 21 and the half cadence that precedes it at mm. 20. In the 18th century, half cadences before the return of A in rounded binary forms is quite common. In the 19th century, however, composers may also elide or otherwise obscure this boundary as Chopin does between mm. 16-17 in the rounded binary form found in mm. 1-24 of his polonaise in A major, Op. 40, No. 1.
In simple binary, there is no substantial return of opening material. The material in the second reprise takes one of two possible manifestations. The first possibility is that it continues with the same sorts of ideas presented in the first reprise (though it is not a repeat of the first reprise) so it would be labeled A′ (note the prime symbol). This type never really leaves the material from reprise 1 so the concept of return is not appropriate because the A material is always present. The second possibility is that the second reprise simply contains relatively new material throughout and so B would be the more appropriate label.
The bourrée from Bach's lute suite in E minor, BWV 996 (Example 3), is a good example of a simple binary form where the second reprise would be labeled A′ because the musical material in the second reprise seems like it is simply continuing the ideas from the first reprise throughout. Notice how there is no clear return of the first reprise's opening material in the middle of the second reprise (therefore, not rounded binary).
Balanced is a term used to describe an aspect of a binary form (either simple or rounded). It means that the tail end of the first reprise, returns at the tail end of the second reprise. That return will be in the piece's home key even if it was in another key in the first reprise. In order to be considered a return, there needs to a crux point, that is a particular moment where the restatement begins at the tail end of the second reprise. This restatement is the point at which there is a direct bar-for-bar mapping of measures between the tail end of both reprises. Importantly, this excludes rounded binary examples where the entire first reprise is repeated verbatim in the second reprise because there is no crux point at the tail end of the second reprise. In the diagram below, the (x) represents the music at the tail end of the first reprise (A section), and its return at the tail end of the second reprise.
In longer simple binary forms, the balancing material can be quite substantial. In Domenico Scarlatti's "sonata" in A major, K. 322 (Example 2), the material that returns is nearly 24 measures long—over half the length of the first reprise—and is easily recognizable by ear. In the Scarlatti work, (x) starts in the middle of m. 21 and ends at the end of the first reprise, m. 44. That material returns in reprise 2 in the middle of measure 58 and continues to the end of the work, with a few new melodic decorations along the way (compare m. 26 and m. 63, for example). Importantly, note that (x) in the 2nd reprise has been transposed back to the home key. When it was stated initially in the first reprise, (x) was in the key of E minor and then E major so it needed to be transposed back to the key of A in order for the work to start and end in the same key.
It is common for each section of the binary form to end with standard cadence types. This is especially true for 18th century classical music but stylistic preferences in the 19th century alter cadential expectations for the first part in particular, sometimes opting for lower levels of closure, ending with tonic prolongational progressions instead of standard cadence types (examples: Schumann, Papillon, 1 (m. 8) & 7 (m. 8), Kinderszenen, no. 9 (m. 8)).
As with other forms, the first reprise of a binary form can be described as harmonically open or closed. The second reprise can be described this way as well, but because binary forms are expected to be monotonal, it usually is implied and not stated explicitly.
If the first reprise of a binary form is open, it may contain a modulation. See the glossary description of standard, classical-era modulation schemes for more information. Regardless of the harmonic situation at the end of the first reprise, you should expect the second reprise to end with an authentic cadence in the original key. There may be additional cadences before the end, but the PAC at the end of the second reprise is essentially an obligatory convention in European-style classical music. See progressive tonality for more information about pieces that start and end in different keys.
As with most aspects of form, binary form moves between relative stability and relative instability throughout the form which serves to give the work a linear drive due to the expectation that a work will start stable, become unstable, and ultimately end with a sense of relative stability. In binary form, you can expect the first reprise to be relatively stable, the beginning of the second reprise to be relatively unstable and the end of the second reprise to return to stability. The return of A material in the second reprise of a rounded binary form is also commonly expected to be a point of relatively stability. The beginning of the second reprise is so often relatively unstable, that some theorists refer to it as a digression or departure (Douglass Green's term), sometimes forgoing the letter B altogether to focus on the function of the music.