When music remains in a new tonal area so long that the ear no longer hears the original tonic as "tonic" any more, the music has modulated. This amount of time varies from person to person, and by the context of each musical passage. For example, some people might hear the following passage as modulating to g minor, while for others it might be heard as only tonicizing g minor (iv).
However, there are a few guidelines that are helpful in determining whether one should analyze a passage as a modulation or a tonicization. Generally, we can say a passage has modulated when one or more of the following is true:
In the above example, the music cadences in g minor, but there is only one chord containing an altered pitch. This passage could be analyzed either as a tonicization or a modulation. The following passage, however, is clearly a modulation, since the altered pitch remains until the final cadence. We call the first chord that contains a pitch from the new key that is not present in the old key as the point of modulation. From this chord on, the passage cannot be analyzed in the old key, so a new series of chord symbols in the new key is used to represent the harmonies.
When a passage changes key abruptly with no commonly functioning chords or pitches, it is referred to as a direct modulation. This type of modulation is often used to create a sudden change of mood via it's sudden change of tonality.
Common Chord/Pivot Chord Modulation
By preceding a modulation with a harmony that functions both in the old and new keys, the change of tonic becomes smoother and less abrupt. A chord that functions in both keys and is the chord immediately before the point of modulation is called a pivot chord. (Some theorists refer to this as a common chord. Either term is acceptable.) Pivot chords are identified by using roman numerals in both keys (one above the other), and a separating line or box to highlight their dual function, as in the following example. One of the previous examples used a pivot chord to smoothly modulate from Bb to gm. Notice how the vi chord in Bb is also the i chord in gm. By using a common pivot chord, our ears may hear the passage as functioning in g minor before the actual point of modulation.
Pivot chords may be found by listing all of the chords in both the old and new keys, rotating the second key's chords so that the same letter names/roots are next to each other. Then identify chords that have both the same root and the same chord quality. Both of these must be the same for a chord to serve as a pivot chord.
For example, let us say we wish to smoothly modulate from D Major to E Major. First, list all of the chords and identify the common chords:
As we see above, we could use an f# minor chord (iii in D Major) or an A Major chord (V in D Major) as a pivot chord. The following two progressions demonstrate how each of these chords could function in this manner:
f#m pivot (iii)
A Major (V) pivot
Pivot Tone Modulations
In works that are not in a chorale style, composers often modulate through the use of a common tone in the melody. By eliminating the harmony, and only sustaining or repeating a pitch that exists in both the old and new keys, the music may smoothly modulate. This technique is frequently found in solo piano and chamber works of the common practice period.
Keys that differ by only a single accidental (i.e. D Major with 2 sharps, and f# minor with 3 sharps) are referred to as closely related keys. Because of their similar key signatures, closely related keys contain multiple pivot chords, including the tonic in each key. This makes a smooth change of key easier for both composers and listeners. Closely related keys are easy to identify using the circle of fifths chart:
All of the keys (both major and minor) that are adjacent a selected key are considered closely related. Notice also that the closely related keys also form all of the non-diminished, diatonic chords in the original key (be sure to use the natural minor scale form for minor keys). Hence, these new tonic's are familiar to the ear, making modulations smoother and more natural sounding.