There are two fundamental forms of modulation - between different tonal centres, and between different tonal types (major and minor tonalities). This means that we can modulate from the major tonality of C to:

• a different tonal centre, but the same tonal type, such as G major;
• the same tonal centre, but a different tonal type, which is c minor;
• a different tonal centre and a different tonal type, such as f minor.

A modulation can be between closely related or a distantly related keys, and it can be articulated in a manner which either smooths this transition or which highlights it as a sudden shift. The choice is the composer's, and in this section I will describe the relationships between the different tonalities and the methods used to move between them.

The distance between keys

There are two important ways to measure the distance between any two keys. The first is the number of notes that they have in common - the more they share, the more similar they are; the second is the inherent similarity of any two keys which share the same tonic note - the only two keys which share the same tonic note are parallel major and minor keys.

Modulating between any two keys which are closely related is likely to be much less disruptive than direct modulation between any two distantly related keys. Of course, it may be desired for the modulation to be heard as a sudden and unexpected transition, in which case no preparation need be made, but if we wish to modulate smoothly to a distantly related key we can do this most effectively by using a series of closely related keys as stepping stones to the more distant key.

The most basic measure of the reltionship between keys is the number of notes they have in common.

The most closely related keys, measured on this basis, are the relative major and minor keys which share all seven notes. For instance, the keys C major and a minor can be said to share all of their notes (if we take the natural minor as the basis of the minor scale, and consider the harmonic minor to be an alteration of it). The relative minor of any major key is a minor third below, the relative major of any minor key is a minor third above, and these keys have exactly the same key signature.

The keys C major, G major, and e minor share six (out of seven) notes - c, d, e, g, a, b. So both G major and its reciprocal F major, and e minor and its reciprocal d minor can be considered to be the next most closely related keys to C major.

The following table shows the number of notes shared by every key and the key of C major (or a minor), so that we can see the most closely related keys at the top and the most distantly related at the bottom:

Minor keys on flat side	Major keys on flat side	No. of notes in common with	Major keys on sharp side	Minor keys on sharp side
		C or a
d	F	6	G	e
g	B	5	D	b
c	E	4	A	f
f	A	3	E	c
b	D	2	B	g
e	G	1	F	d

Modulation between any of these closely related keys is easy to do, and can be achieved quickly and simply.

The most common method of making such a modulation as smooth as possible is to use a pivot chord, which must be a chord that is found in both keys. The pivot chord is approached as a member of the original key but then quitted as a member of the new key which is established with a cadential progression. For instance to modulate from C major to e minor, we could use the progression

C - F - G - a - B - e

In this context, a is the pivot chord - it is approached as vi of C major but quitted as iv, in the cadential progression iv - V - i, of e minor.

The above table is, however, something of a simplification because it ignores the perceived similarity betweem parallel major and minor keys. For instance, the keys of C major and c minor are heard as similar, despite the fact that they share only four notes, because they share the same tonic note (c) as well as the same dominant chord (G7).

The easiest way to modulate between parallel keys is simply to use the common dominant chord as the pivot:

C - F - G - c - f - G - C etc.

Although a simple transition, without any pivot can also be effective.

The integration of these two types of similarity - keys with many shared notes and parallel keys - can be most effectively illustrated by using the chart of the regions invented (discovered) by Arnold Schoenberg, and first published in his Structural Functions Of Harmony in 1954. By regions, Schoenberg is referring to the tonal areas through which a piece of music journeys - he used this term to indicate that most of the tonalities in a piece of music can be heard in reference not just to their preceding and proceeding tonalities, but also to the central tonality of the piece (usually its starting and ending tonality), and this concept is known as monotonality. I reproduce an interpretation of Shoenberg's chart below with some modifications to make it more easily understandable.

In these charts all closely related keys are contiguous, such that keys which share six notes are found above, below or diagonally, while major and minor parallels and relatives are found to the left or right. Also, the closest relationships to C and a, in their respective charts, have been highlighted.

The charts can be used as a tool to navigate smoothly between otherwise distantly related keys by using each of these contiguous steps as stepping stones. For instance, to modulate smoothly from a minor to the distantly related g minor (they have only two common notes), we can modulate from a to A to c to g - each of these individual steps is easy to achieve and together they can quickly link the two keys.

Of course, if one wishes, it is possible to take a longer route between any two keys, for instance from a to D to e to E to B to F to d to g. But it is important to remember that there is nothing inherently superior about taking the shortest possible route, and in most circumstances, when modulating to a distant key, a longer path is more effective beacuse it gives the ear time to adjust.

The final method of modulation is achieved through using a chord which can be enharmonically reinterpreted to enable it to function as a pivot chord between two keys. There are a limited number of chords which are capable of being enharmonically reinterpreted:

• The dominant seventh (1 - 3 - (5) - 7), which can be reinterptreted as an augmented sixth (1 - 3 - (5) - 6), and vice versa.
• The augmented triad (1 - 3 - 5), which can be reinterpreted as the augmented triad built a major third above or below (i.e. (a - c - e) = (c - e - g) = (e - g - b).
• The diminished seventh (1 - 3 - 5 - 7), which can be reinterpreted as a dimished seventh built a minor third or tritone above or below (i.e. (e - g - b - d) = (g - b - d - f) = (b - d - f - a) = (d - f - a - c) = (f - a - c - e).

By approaching the chord according to one interpretation and quitting according the other, quite radical modulations can be quickly undertaken. For instance: the dominant seventh G⁷ can be approached as V⁷ of C, but instead quitted as VIaug6 of B; the augmented triad E+ can be approached as III+ of c minor, but quitted as V+ of e minor; and the diminished seventh b⁰⁷ can be approached as vii⁰⁷ of C major but quitted as vii⁰⁷ of G major.

Enharmonic modulation can be used to jump horizontally from one to four steps on the chart of the regions, and so is capable of extravagant tonal shifts that would be otherwise absurd; but just because a modulation is possible does not mean it is good. Enharmonic modulation is a sophisticated and somewhat deceptive device, and it must be handled with care - too much use of enharmonic reinterpretation can sound wearing to the ear and is unaesthetic, but occasional use can provide modulations of unparalleled excitement and suprise.