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''Modulation'' is a the practice of changing keys inside of a composition.  This can be done for a variety of reasons, but its basic purpose is to contrast or create tension against the previous (and possibly even prior) key(s).  Typical modulations are found when themes or sections change.  They can also be used to seemingly add energy to a prior theme/section.

== PreReqs ==

== Common Modulations ==
Modulations usually take place between highly-related keys (those with the most common notes).  See the table below.

{|class="wikitable"
!colspan="2"|
!colspan="3"| Key 2 Type
|-
! Key 1 Type
! Key 2 Root's<br>relation to<br>Key 1 Root
! Major !! Minor !! Harmonic<br>Minor
|-
!rowspan="13"| Major
|-
| '''U''' ||  || 4 || '''5'''
|-
| m2 || 2 || 3 || 3
|-
| '''M2''' || 5 || '''6''' || '''5'''
|-
| m3 || 4 || 2 || 3
|-
| '''M3''' || 3 || '''6''' || '''5'''
|-
| '''P4''' || '''6''' || 3 || 3
|-
| tt || 2 || 4 || 4
|-
| '''P5''' || '''6''' || 5 || 3
|-
| m6 || 3 || 2 || 3
|-
| '''M6''' || 4 || '''7''' || '''6'''
|-
| m7 || 5 || 2 || 4
|-
| '''M7''' || 2 || 5 || 4
|-
!rowspan="13"| Minor
|-
| '''U'''  || 4 ||   || '''6'''
|-                
| m2 || 5 || 2 || 4
|-                
| '''M2''' || 2 || 5 || 4
|-
| '''m3''' || '''7''' || 4 || '''5'''
|-                
| M3 || 2 || 3 || 3
|-                
| '''P4''' || 5 || '''6''' || '''5'''
|-                
| tt || 4 || 2 || 3
|-                
| '''P5''' || 3 || '''6''' || '''5'''
|-                
| '''m6''' || '''6''' || 3 || 3
|-                
| M6 || 2 || 4 || 4
|-                
| '''m7''' || '''6''' || 5 || 3
|-                
| M7 || 3 || 2 || 3
|-
!rowspan="13"| Harmonic Minor
|-
| '''U''' || 4 || '''6''' ||  
|-
| m2 || 4 || 3 || 3
|-
| '''M2''' || 4 || 3 || 3
|-
| '''m3''' || '''6''' || 4 || 4
|-
| M3 || 3 || 3 || 4
|-
| '''P4''' || 3 || '''5''' || 4
|-
| tt || 4 || 3 || 4
|-
| '''P5''' || 3 || '''5''' || 4
|-
| '''m6''' || '''5''' || 3 || 4
|-
| M6 || 3 || 4 || 4
|-
| m7 || '''5''' || 4 || 3
|-
| '''M7''' || 3 || 3 || 3
|}

This means the most highly-related keys are the relative major/minor keys, which have the exact same set of notes but different roots (C Major and A Minor).  For any major key, the most related major keys are those whose root is the 4th or 5th note of the original key.  The most related minor keys (other than the relative minor) are those based on the 2nd and 3nd notes in the major key.  Similarly, for a minor key, its most related minor keys are based on its 4th and 5th notes, while its most related major keys are based on the 6th and 7th notes (inverse of 2nd and 3rd).

Harmonic minor's raised 7th causes it to be less related to other keys of all variations covered above; however, it basically shares the same relationships to major and minor keys as its natural minor variation.  The most related major keys are based on the 6th and {{f}}7th notes.  The most related minor keys are based on the 4th and 5th.  Interestingly, almost all modulations to other harmonic minor keys are equally inter-related.

=== Examples ===
* '''I => IV''' - (C => F, F => Bb, G => C)
* '''I => V''' - (G => D, C => G, A => E)
* '''i => III''' - (Am => C, Em => G, B{{f}}m => D)
* '''i => {{s}}iv''' - (Ahm => D{{s}}hm)

== Modulation Methods ==
=== Common Chord ===
Uses a chord common to the source and destination keys to transition between them.

=== Enharmonic Chord ===
Even if two keys do not have the same chords in way of naming conventions, they may have chords that are enharmonically equivalent.  For example, the keys of C{{s}} and C{{f}} have no chords in common by name.  C{{s}} has only {{s}} chords and C{{f}} has only {{f}} chords.  Yet, E{{f}}m and D{{s}}m are enharmonic, as are G{{b}} and F{{s}}.

=== Common Tone ===
A note common to the source and destination key is sustained during the transition.  For example, a C ==> Em modulation could sustain the E in the I chord of C while moving to the i chord of Em, which also contains E.

=== Chromatic ===
Useful for modulating between two keys that are not closely related, chromatic modulation uses a chromatic line to bridge the keys.  In other words, for the chords chosen, one voice can move up or down 2 or more half-steps.  This necessarily implies the transition will occur over at least 3 chords.  For the strongest effect, the chromatic line should lead into the root of the new key.  

For example, consider a modulation from C Major to A Harmonic Minor.  C Major has a G note, while A Harmonic Minor has G{{s}} and A.  This allows us to create a G - G{{s}} - A chromatic line through the following progression:  G - E - A.  That's ('''G''' B D) - (E '''G{{s}}''' B) - ('''A''' C E).  If we follow good [[Voice Leading|voice leading]], we can keep the G, G{{s}}, and A notes in the same voice, thus getting the chromatic line.  G is the V chord of C Major.  E is the V chord of A Harmonic Minor.  The progression could be written as:

    G       E       Am
 C: V  Ahm: V       i
 E--3-------0-------0------
 B--0-------0-------1------
 '''''G--0-------1-------2------'''''
 D--0-------2-------2------
 A--2-------2-------0------
 E--3-------0--------------

<small>''Notice the chromatic line on the G string.''</small>

Depending on the keys used, it may be more difficult to achieve the chromatic line with only the diatonic chords of the two keys.  In these cases, we can resort to using a [[Borrowing Chords/Notes#Secondary Dominant|secondary dominant]] or some other non-diatonic chord.

=== Phrase ===
A phrase modulation avoids any transition whatsoever, occurring abruptly between two virtually independent phrases.  The first phrase in the original key will end on a cadence, finding resolution and ending.  The next phrase begins in the new key, with no harmonic or melodic bridge to ease the transition.  This modulation is commonly between closely-related keys, preventing a completely disjointed feel.  However, the transition is typically quite noticeable, which is the point of this modulation method.

=== Sequential/Rosalia ===
A phrase or progression ending in a cadence can be transposed up or down to a different key and repeated, once or several times.  Here the harmonic function of the sequence of chords is less important than the effect of the sequential motion.  The transpositions may follow the diatonic notes of the original key, or may move completely chromatically.  The sequence can repeat enough times to return to the original key, or be used to arrive at a new key.

For example, consider the following progression: 

     Cm      B          B{{f}}m     A          A{{f}}m  G     E{{f}}m   A{{f}}m
 Cm: i       VII  B{{f}}m:  i       VII  A{{f}}m:  i    VII   v     i

=== Chain ===

=== Parallel ===

== Homework ==

== Q & A ==

== Related Lessons ==
* [[Voice Leading|Prev: Voice Leading]]
* [[Borrowing Chords/Notes|Next: Borrowing Chords/Notes]]