Of these chords, V (G major) is said to be the dominant of C major (the dominant of any chord is the one whose root is a fifth higher). However, each of the chords from II through VI also has its own dominant. For example, VI (A minor) has E major as its dominant. These extra dominant chords are not part of the key of C major as such because they include notes that are not part of the C major scale. Instead, they are the secondary dominants.
Below is an illustration of the secondary dominant chords for C major. Each chord is accompanied by its standard number in harmonic notation. In this notation, a secondary dominant is usually labeled with the formula "V of ..."; thus "V of II" stands for the dominant of the II chord, "V of III" for the dominant of III, and so on. A shorter notation, used below, is "V/II", "V/III", etc. The secondary dominants are connected with lines to their corresponding tonic chords.
Note that of the above, V/IV is the same as I. However, as will become clear shortly, they are not always identical.
Like most chords, secondary dominants can be classified by whether they contain certain additional notes outside the basic triad; for details, see Figured bass. A dominant seventh chord (notation: V7) is one that contains the note that is a minor seventh above the root, and a dominant ninth chord (notation: V9) contains the note a ninth above the root. For instance, V7/IV, although it is a C chord, is distinct from regular C major because it also contains the note B flat, which is a minor seventh above the root of C, and not part of the C major scale.
To illustrate, here are the secondary dominants of C major, given as dominant seventh chords. They are shown leading into their respective tonics, as given in the second inversion.
Secondary dominants are normally major chords, not minor; thus, the secondary dominant of VI in C major is considered to be E major, not E minor. This accords with the fact that in a minor key, it is normally the dominant major chord, not the dominant minor, that might serve the function of dominant harmony.
When used in music, a secondary dominant is very often (though not inevitably) directly followed by the chord of which it is the dominant. Thus V/II is normally followed by II, V/VI by VI, and so on. This is similar to the general pattern of music wherein the simple chord V is often followed by I. The tonic is said to "resolve" the slight dissonance created by the dominant. Indeed, the sequence V/X + X, where X is some basic chord, is thought of by some musicians as a tiny modulation, acting as a miniature dominant-tonic sequence in the key of X.
The concept of the secondary dominant was not recognized in writings on music theory prior to 1939. Before this time, in music of Bach, Mozart, Beethoven, and Brahms, a secondary dominant, along with its chord of resolution, was considered to be a modulation. Because the effect of modulation was so short, and did not sound like a real arrival of a new key, the two chords had a special name--"transient modulation"--that is, a modulation in which the new key is not established. Since this was a rather self-contradictory description, theorists in the early 1900s, such as Frank Shepard, Benjamin Cutter, and George Wedge, searched for a better description of the phenomenon.
In 1939, in a monograph entitled "Principles of Harmonic Analysis," Walter Piston first used the analysis "V7 of IV." (Notably, Piston's analytical symbol always used the word "of"--e.g. "V7 of IV" rather than the virgule "V7/IV.) In his 1941 "Harmony" Piston used the term "secondary dominant" for the first time. It has been generally accepted in music theory since then.
In the Fifth edition of "Harmony" by Walter Piston and Mark DeVoto, a passage from the last movement of Mozart's Piano Sonata K. 283 in G major serves as one illustration of secondary dominants. Below, the harmony alone is first given, labeled both for the literal names of the chords and for their chord number in the key of G major.
It can be seen that this passage has three secondary dominants, each one followed (as expected) by the chord of which it is the dominant. At the end, there is a standard dominant-tonic cadence, which concludes the phrase. The lines drawn below the diagram show each instance in which a dominant is followed by its corresponding tonic.
The harmony is distributed more subtly between the notes, and goes faster, in Mozart's original:
The secondary dominants here create a rapidly descending chromatic harmony, an effective lead-up to the tonic cadence at the end of the phrase. There are many similar passages in Mozart's music.