Modulation

To study modulation, simply put, is to study the various keys through which a piece of music passes. While there are numerous issues, here are three of the most important things to consider: 1) how long is the music in a particular key; 2) what are the relationships between the keys; and 3) how did the music get from one key to the other?

A word about figuring out what key the music is in: while there are numerous things to think about (most especially accidentals), here is a maxim that will work virtually all of the time:

Bass line patterns and cadences show the keys

How long is it in the key?

Here is an approximate hierarchy of chromaticism--that is, a range from music that's totally diatonic to music that's so chromatic that it's very difficult to get a handle on the tonality. Some of the chord types may be unfamiliar to you right now; that's OK. New terms which refer specifically to the length of time in a key are in bold.

Functional use of diatonic chords within one tonality; no chromaticism at all
Addition of chromatic nonchord tones in the melody (not yet chromatic harmony)
Chromatic harmony:
- mode mixture--borrowing from parallel scale--iv, iiø65, ^bVI in a major key, I in minor, etc.
- secondary chords--secondary dominants, secondary diminished sevenths, and so on
- altered chords--Neapolitan, augmented sixth chords
- nonfunctional or melodically derived chords
Tonal region or tonal cell--a short group of chords, often including a cadence, that very temporarily establishes a new tonic. This is when we've said "oooooooof" something in class
Change to new key area, established by at least two phrases in the new key, with eventual return to the original tonality. The new key area can be of varying length
Change to new key area, established by at least two phrases in the new tonality, with no return to original tonality
Tonal instability--continual, often rapid changes of key

Key relationships

Parallel keys: scales with the same tonics. This is really called change of mode or mutation rather than change of key--examples: C major and c minor, D major and d minor
Nearly-related keys (also called closely-related keys): keys with the same key signature or with one accidental more or less in the key signature. These are keys that are adjacent on the circle of fifths. Examples: C major is nearly-related to d minor, e minor, F major, G major, and a minor. C minor is nearly-related to E^b major, f minor, g minor, A^b major, and B^b major. A quick rule of thumb: for the nearly-related keys to a major key, think ii iii IV V vi. For the nearly-related keys to a minor key, think III iv v VI bVII. Note that nearly-related keys include relative major and minor keys (for example, C major and a minor or e minor and G major).
Distantly-related keys: keys that aren't nearly-related (key signatures over one accidental apart). A special type is the chromatic third relationship: two keys a major or minor third apart, where the tonic triads of the two keys have only one or no notes in common. Example: C major is in a chromatic third relationship with E^b major, e^b minor, E major, A major, A^b major, and a^b minor. This is more common in 19th-century music.

How did the music get from one key to the other?

Common chord--also called pivot chord--a (usually diatonic) triad or seventh chord which is exactly the same notes but has a different Roman numeral function in both keys. Example: the chord A C E is a vi chord in C major, but it's a ii chord in G Major. Common chords are often a pre-dominant chord in the new key. Often happens within a phrase.
Tonal shift--also called phrase modulation--usually takes place between phrases; some theorists don't consider it to be a true modulation because it isn't really a process.
Chromatic inflection--often distinct texturally from common chord. Usually the introduction of a new melodic leading tone; often it's related to the abrupt introduction of the dominant chord of the new key. May occur within or between phrases.
Common tone--using a single note that two keys have in common (often connects chromatic third related keys). Often between phrases. Often involves the tonic triads and scale degrees 1, 3, or 5 in the two keys. More common in 19th-century music.
Modulating sequence--some harmonic pattern that is repeated in a sequential manner in several different keys; usually between phrases, though the patterns may be connected via common chords. Very common in Baroque music.
Enharmonic modulation--can include enharmonic keys, chords, or just notes.

Remember that major keys will often modulate to the dominant, minor keys to the relative major. Remember, too, that you won't see a new key signature every time the key changes.

Writing: the important thing is to establish the keys through stock progressions such as I V43 I6 IV I64 V7 I; I vi IV ii V7 I; etc. These progressions may of course be embellished with secondary dominants. So, you establish the original key, make the connection (say, with a common chord), then establish (or confirm) the new key.

Hearing: THINK about potential key relationships. Listen to the intervallic relationship between the keys; listen to the mode (major or minor) of the new key; focus especially on the area where the actual modulation occurs. Hearing modulation is often a retrospective process--after it happens you say, "hey--we're in a new key--how did it get there?"

Another way of thinking about it

A harmonic link between the keys (common chord, etc.). Here, the vi chord in D major becomes a ii chord in A major. Note how the first key is clearly established, then "the action" takes place, then the second key also is clearly established. The common chord is a pre-dominant chord in the new key; this is most typical.

      D:  I  IV I64  V    I   vi
                            A:ii I64 V    I   IV I64  V    I

Melodic links between the keys:

Chromatic inflection: here, a simple common chord interpretation doesn't satisfactorily explain the change of key. The initial key of B^b major is established, then the new leading tone in the new key of g minor (the F#) is introduced in the bass voice. The key of g minor is then established.

      Bb: I   I6   V  g:V6   i   iv6   V4-3     i

Common tone: here, there simply isn't a common chord, and it's pretty obvious that the pitch A in the melody is what connects the initial key, F major, with the new key of D major. The pitch A is scale degree 3 in F major, then it becomes scale degree 5 in D major. Notice that these are chromatic third related keys--this is most typically the case when dealing with a common tone modulation.

   F:   I   IV  I64  V   I      D: I   IV  I64  V   I

A formal link between the keys (tonal shift or phrase modulation, modulating sequence, etc.). Here, Schubert elects to shift directly from d minor (mm. 1-4) to a minor (mm. 5-8). It happens between the two phrases, and while a common chord or even common tone interpretation is theoretically possible, the change between phrases and the dramatic changes in dynamics, range, and texture all point to a rather direct tonal shift from d minor to a minor.

    d: i        V7  i V7 i a: i            V7       i