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Modulation
Modulation is the musical act of changing from one key to another. Often modulations occur smoothly, that is to say, a certain amount of musical time is taken to effect the transition and the new key is closely related to the original one so that the ear is not jarred significantly. But sometimes an abrupt change of key may occur without any warning, or possibly the new key will be distant from the original one so that the musical effect is very striking.
After the section in the new key, and possibly additional excursions to other keys, most pieces return to end in the original key. Often the tonal structure of the work will be part of its formal structure, such as a piece in sonata form, where the appearance of the second theme is usually identified with the point of arrival of the modulation to the new key.
Common chord modulation
The common chord modulation is the most common type of modulation, where a chord in the original key also functions as a chord in the new key, and is used at the moment of transition from one key to the other. This chord with dual identities is called the pivot chord. Often the common chord is diatonic, but chromatic common chords can also be used, and are especially useful when the composer wishes to effect a modulation to a more distant tonality.
There are three things that have to occur to have a good common chord modulation:
1. The original key must be clearly established for the listener
2. A pivot chord, which is analyzed in both keys, occurs as a transition from the original key to the new key. Sometimes there are two or more chords at this transition point that belong to both keys instead of just one.
3. The new key must be clearly established
A typical simple example might be a modulation from a major tonic to the key of the dominant, for example, D major to A major. Since these two keys differ by only one note (the note G natural in D major, G sharp in A major) there are four diatonic triads that are common to both keys: D major, f# major, A major, and b minor. Any one of these could serve as the pivot chord in this modulation.
In order to clearly establish the key of D major, one or more chords that contain the note G natural should be used before the pivot chord occurs, such as ii or IV. If there are no chords containing the note G natural, all of the diatonic triads in the original key area could also be triads in the new key of A major. Therefore the entire progression could be analyzed in the new key, and there would be no modulation. Likewise, after the pivot chord occurs, in order to clearly arrive in the new key, the G sharp must occur. This is not difficult in this example because G sharp is the leading tone of A, and is present in the dominant triad.
An example of such a modulation follows:
D maj: I V6 I ii6 V7 vi
A maj: ii V I V6 I
The progression begins in D major. The pivot chord is the sixth chord in the progression, the b minor triad. This chord functions as vi in D major and ii in A major. It is analyzed in both keys, but after that chord occurs, the analysis switches to the key of A major.
Key relationships
The number and type of chords that may be common to the two keys that you wish to link by modulation depends on their distance from each other on the circle of fifths. These fall into three groups:
1. The close keys which share common diatonic triads. This includes closely related keys and their next-door cousins, keys that are no more than two notches away on the circle of fifths.
2. Chromatic mediant keys, keys which are major or minor thirds above and below the original tonic, and are both major or both minor keys. These keys are three and four notches away on the circle of fifths.
3. Distant keys, both major or both minor, which are a half step up or down or a tritone away from the starting key. These are five or six notches away on the circle of fifths.
The term closely-related keys is defined as keys which have no more than one sharp or flat different in their key signatures. This term also includes the relative major or minor of the original key (same key signature). Relative keys have all their triads in common. From the original key, the keys located one notch away on the circle are always a perfect fifth up or down from the original key, the keys of the dominant and the sub-dominant, including the relative minors of those three keys (or the relative majors if original key was minor). Keys that are one notch away from each other will always have four triads in common, since there is only one note different in the key signature. That one note affects three of the seven triads, leaving the other four as common chords.
If you go two notches away from your original key on the circle of fifths, you will find that the keys there will be one whole step up or down from your original tonic. When the triads in these keys are compared with the original, you will find that there are still two triads that they share in common. For example, if you look for common chords between D major and E major, you will find that the F-sharp minor triad and the A major triad are common to both keys. Likewise, if you went two notches in the other direction on the circle, from D major to C major, you would find that these two keys have the E minor and G major triads in common.
This means that all keys no more than two notches away in either direction on the circle of fifths may be linked with a common diatonic triad. Therefore, a common chord modulation will work well for these types of modulations, by using a triad which is diatonic in both keys for the pivot chord. For example, from the key of D major, one can modulate to the keys of G, A, C, and E major, and b, e, f#, a, and c# minor using a diatonic triad as a pivot chord.
Chromatic mediant keys are keys whose tonic notes are a major or minor third apart, and which are both major or both minor keys. Some examples of chromatic mediant relationships are C major and A-flat major, F major and D major, or B major and E-flat major (enharmonically). Since these keys are more distant than those discussed above, they share no common diatonic triads at all. When trying to make a modulation of this distance, it is often useful to link the two keys through their parallel major or minor. This usually involves using mode mixture, where a diatonic triad is “borrowed” from the parallel minor of the original or new key to serve as the pivot chord.
For example, to make a modulation from A major to F major, one could borrow a triad from the parallel minor of the original key, A minor, to use as the pivot chord. Notice that A minor is only one notch away from F major on the circle of fifths, and therefore it shares four common triads with F major. For one example, one could “borrow” the sub-dominant chord from A minor (the d minor triad), and use it in A major in place of the regular IV chord (a D minor triad instead of the usual D major triad). Even though this chord is chromatic in the key of A major, it still sounds quite acceptable, since it is related to the same tonic note, A. In this example, the D minor chord functions as the iv in A major, and the vi in F major.
Another example might be the modulation from C major to E major. In this example, looking for a chord to borrow from the parallel minor of C, brings us over to a key signature of three flats. This is even further away on the circle from E major than the original key. The secret here is to borrow from the parallel minor of the second key – the key of e minor – which is much closer to C major, only one notch away on the circle. This means that there are four triads that could be borrowed from E minor and used in E major that would work as pivot chords to link us to C major. Here are two examples: the minor tonic triad in E would be iii in C major, the major VI triad in E minor would be I in C. This modulation would use a chord that was chromatic in the second key (borrowed from the parallel minor) to make the link.
Distantly-related keys are those whose tonic notes are a half step away from each other (five notches away on the circle of fifths), or a tritone away(six notches), and both keys are major or both minor. These keys share very few things in common. C major and B major, for example, only share two notes in common, B and E. The keys of C and F-sharp major share one note, but two pitches in common: both use the note B, and the other note is an enharmonic equivalent: E#/F. These keys do not share enough tones in common to make any sort of triad. This means that one usually has to use some sort of “trick” chord, which gets respelled in the second key, to link them together.
For example, one might modulate a tritone using a fully diminished seventh chord built on the leading tone, vii07. This chord, when respelled, also functions as the leading tone chord in a new key a tritone away. This is possible because of the structure of this type of chord: when it is played, out of context, the listener cannot tell which note is the root. Through different spellings, any note can be made to function as the leading tone.
By using this device, the leading tone chord in D minor, C# fully-diminished 7 (C#, E, G, B-flat), could be used to modulate to A-flat minor, if it were respelled as G, B-flat, D-flat, F-flat. This chord would function as vii07 (fully-diminished) in both keys, although the inversion numbers would be different in the musical analysis, since the existing bass note would not be the same part of the chord in the new key.
This key relationship is very distant and hinges on this one chord; therefore a modulation like this cannot be done quickly, unless the shocking resolution of the chord is a desired effect. Often in a situation of this type, the time spent on the diminished chord will be quite prolonged, to give time for the ear to get lost a bit in the ambiguity of the diminished harmony, and thus the unusual resolution seems somewhat less surprising.
A chromatic chord that can be used as a pivot to link keys a half step apart is the German augmented 6th chord. This chord, although spelled differently, sounds exactly like a dominant 7th chord, and therefore the two functions can be interchanged.
For example, the German augmented 6th chord in C (A-flat, C, E-flat/D#, F#) could be used to become the V7 in the key of D-flat major (respelled as A-flat, C, E-flat, G-flat). Or the V7 chord in any key could be respelled to become the German 6th chord in another key.
If a modulation that occurs like this begins in a key with many sharps or flats, sometimes there will be an enharmonic key signature change from sharps to flats, or vice versa to get to the new key. For example, if the original key was D-flat major, and the German 6th were used as the pivot chord, it would be easier to change enharmonically to sharps for the second key. The German 6th (spelled as B-double-flat, D-flat, E natural, G natural) would then become the V7 of E-double-flat major. Since there is no key of E-double-flat on the circle of fifths, it would make a lot more sense to change this new key enharmonically to D major. Therefore this modulation would move from a key signature of five flats to one with two sharps.
In general, only rarely in a composition is such a chord spelled in the original key and then respelled in the new key. More likely, such an enharmonic pivot chord will be spelled in the manner it resolves, or in other words, in the way it functions/resolves in the second key. In the last example, the German 6th might only be spelled as the V7 of D major, and it is incumbent upon the listener to hear the chord’s function (the German 6th) in the original key, and its relationship to the new key, and for the musical analysis to label it correctly.
These sorts of modulations are very difficult to think through, and even once the pivot chords are spelled correctly and understood, attention to smooth voice leading and good harmonic rhythm will help make this distant relation more believable to the ear.
Chord qualities for pivot chords
Since both diatonic and chromatic chords can be used as pivot chords in a common chord modulation, the more numerous the possible functions (Roman numeral analysis) for a given type of triad quality, the more useful that type of triad will be as a pivot chord. For example, major triads have more potential functions than any other kind of triad. They can be diatonic triads in major and minor keys, they can be “borrowed” chords when using mode mixture, they can be secondary dominant triads, and they can be Neapolitan chords.
Minor triads, since they cannot be secondary chords, have only diatonic and borrowed functions. Diminished triads have even fewer diatonic options, since there are only two common places where they occur in a scale; augmented triads are very not useful at all as pivot chords, since the augmented triad does not have any diatonic function in a major scale.
Seventh chords, both diatonic and chromatic, can also be used as pivot chords. However, these are less commonly used than the diatonic triads.
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