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The scheme of diatonic major and minor key modulations in the natural keys

© Miroslaw Majchrzak
mmajchrzak77@wp.pl


The circle of fifths is a set of keys arranged in the order of fifths. Let us mark the keys(1) in the circle of fifths with the consecutive integers: the sharp keys with positive numbers, the flat keys -- with negative numbers. The absolute value of the integer designates the number of accidentals in the key.

Example:
The number (4) marks the keys of E major and C sharp minor; the number (- 2) -- the keys of B flat major and G minor.


The set of keys containing all the sounds

We can precisely specify for any given note which keys it can be found in. For instance, the sound C appears in these keys: 1, 0, -1, -2, -3, -4 and -5.
Can we find a major key containing the sound Dx (D double sharp), without changing this note enharmonically into E?
Moving left or right in a series of fifths {F double flat, ..., E flat, B flat, F, C, G, D, A, E..., B double sharp} we will reach sounds that appear in very remote keys with a very large number of accidentals in the key signature.
The successive columns in the table below represent:
A -- notes;
B -- the keys in which those notes appear.

Table 1
AB
B double sharp 20, 19, 18, 17, 16,15, 14
E double sharp 19, 18, 17, 16,15, 14, 13
A double sharp 18, 17, 16,15, 14, 13, 12
D double sharp 17, 16,15, 14, 13, 12, 11
G double sharp 16,15, 14, 13, 12, 11, 10
C double sharp 15, 14, 13, 12, 11, 10, 9
F double sharp 14, 13, 12, 11, 10, 9, 8
B sharp 13, 12, 11, 10, 9, 8, 7
E sharp 12,11, 10, 9, 8,7, 6
A sharp 11, 10, 9, 8,7, 6, 5
D sharp 10, 9, 8,7, 6, 5, 4
G sharp 9, 8,7, 6, 5, 4, 3
C sharp 8, 7, 6, 5, 4, 3, 2
F sharp 7, 6, 5, 4, 3, 2, 1
B 6, 5, 4, 3, 2, 1, 0
E 5, 4, 3, 2, 1, 0, -1
A 4, 3, 2, 1, 0, -1, -2
D 3, 2, 1, 0, -1, -2, -3
G 2, 1, 0, -1, -2, -3, -4
C 1, 0, -1, -2, -3, -4, -5
F 0, -1, -2, -3, -4, -5, -6
B flat -1, -2, -3, -4, -5, -6, -7
E flat -2, -3, -4, -5, -6, -7, -8
A flat -3, -4, -5, -6, -7, -8, -9
D flat -4, -5, -6, -7, -8, -9, -10
G flat -5, -6, -7, -8, -9, -10, -11
C flat -6, -7, -8, -9, -10, -11, -12
F flat -7, -8, -9, -10, -11, -12, -13
B double flat -8, -9, -10, -11, -12, -13, -14
E double flat -9, -10, -11, -12, -13, -14, -15
A double flat -10, -11, -12, -13, -14, -15, -16
D double flat -11, -12, -13, -14, -15, -16, -17
G double flat -12, -13, -14, -15, -16, -17, -18
C double flat -13, -14, -15, -16, -17, -18, -19
F double flat -14, -15, -16, -17, -18, -19, -20

Let us mark the notes arranged in a series of fifths by analogy to the keys -- with integers. We will mark the sound C with the number 0, and the consecutive notes in the series of fifths -- with the consecutive positive integers (up from C) and the consecutive negative integers (down from C).

Example:

Table 2
Notes F flat G flat C F sharp E sharp E double sharp
Note markings -8 -6 0 6 11 18

The set of notes in key 1 is as follows: {1, 3, 5, 0, 2, 4, 6}. Observing the principles of enharmonic change, let us transpose the set of notes in key 1 to key 19. The set of notes in key 19 is as follows: {19, 21, 23, 18, 20, 22, 24}. The highest pitch is 24. The note 24 can be found in the following keys: {25, 24, 23, 22, 21, 20, 19}. The keys 25 and 23 cannot be found in table 1. Similarly, moving down the circle of fifths, we will eventually reach keys which do not appear in table 1.
By analogy, we can progress towards +8 (upwards) and -8 (downwards) around the circle of fifths. The infinite set of all keys can be illustrated by means of the following function:

Table 3

where:
1) the axis y -- are keys with a fixed number of black keys on the piano keyboard(2);
2) the axis x -- is a segment of the set of keys (from 1 to 50) on the scale from minus to plus infinity.
Enharmonically corresponding keys:
1) -∞, ..., 0, 12, 24, 36, 48, 60, ..., +∞
2) -∞, ..., 9, 21, 33, 45, ..., +∞
3) -∞, ..., 4, 16, 28, 40, ..., +∞


Diatonic modulation

To illustrate the possibility of moving from one key to another in the process of modulation, we can present all the major and minor keys or chords in one scheme in the form of a chessboard. The major keys and chords fit into the black fields, while the minor keys and minor tonics -- into the white fields.

1) If the numerical symbols inscribed into the fields represent the major and minor natural keys, we can observe the following qualities(3):
a. Keys in the neighbouring fields have four chords in common, and these chords may be used in the process of modulation between these keys;
b. Keys not adjacent to each other, but separated by only one key, have two chords in common which may be used in the process of modulation between these keys;
c. Keys which are three fields apart have no common chords formed on the degrees of these keys.
2) If we treat the symbols inscribed into the fields as chords, then:
a. Chords in the neighbouring fields may be used to modulate from one key to another;
b. Chords which are two or more fields apart cannot be directly used to modulate from one key to another (we do not take ‘spread out’ horizontal chords into account at this point).
The possibilities of diatonic modulation between all the keys in the tonal space are represented in the "chessboard" scheme below:

Table 4
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
... ... ... ... ... ... ... ...
12 12 12 12 12 12 12 12
11 11 11 11 11 11 11 11
10 10 10 10 10 10 10 10
B sharp minor D sharp major B sharp minor D sharp major B sharp minor D sharp major B sharp minor D sharp major
G sharp major E sharp minor G sharp major E sharp minor G sharp major E sharp minor G sharp major E sharp minor
A sharp minor C sharp major A sharp minor C sharp major A sharp minor C sharp major A sharp minor C sharp major
F sharp major D sharp minor F sharp major D sharp minor F sharp major D sharp minor F sharp major D sharp minor
G sharp minor B major G sharp minor B major G sharp minor B major G sharp minor B major
E major C sharp minor E major C sharp minor E major C sharp minor E major C sharp minor
F sharp minor A major F sharp minor A major F sharp minor A major F sharp minor A major
D major B flat minor D major B flat minor D major B flat minor D major B flat minor
E minor G major E minor G major E minor G major E minor G major
C major A minor C major A minor C major A minor C major A minor
D minor F major D minor F major D minor F major D minor F major
B flat major G minor B flat major G minor B flat major G minor B flat major G minor
C minor E flat major C minor E flat major C minor E flat major C minor E flat major
A flat major F minor A flat major F minor A flat major F minor A flat major F minor
B flat minor D flat major B flat minor D flat major B flat minor D flat major B flat minor D flat major
G flat major E flat minor G flat major E flat minor G flat major E flat minor G flat major E flat minor
A flat minor C flat major A flat minor C flat major A flat minor C flat major A flat minor C flat major
F flat major D flat minor F flat major D flat minor F flat major D flat minor F flat major D flat minor
-9 -9 -9 -9 -9 -9 -9 -9
-10 -10 -10 -10 -10 -10 -10 -10
-11 -11 -11 -11 -11 -11 -11 -11
-12 -12 -12 -12 -12 -12 -12 -12
... ... ... ... ... ... ... ...
-∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞


Examples:
1) Modulating from the G major key to F sharp minor, we can use D major and B minor as the transition chords.
2) Modulating from the A flat major to D minor, we may pass through the chords: C minor and B flat major, C minor and G minor, E flat major and G minor, E flat major and B flat major.
3) Direct modulation between B flat major and G major, D flat major and G minor, F major and B minor, etc., is impossible.
4) We will find common chords in the (direct) modulation between the keys of A flat major and C minor, A minor and G major, B minor and E major.
5) The A major chord appears in the following keys: D major, B minor, F sharp minor, E major and C sharp minor.


Translated by Tomasz Zymer
1. All the reflections in this paper concern exclusively major and minor natural keys.
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2. The sharp keys and the flat keys have a different order of white and black keys, for instance, the B flat major key and the D major key.
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3. We do not take into consideration the keys located on the same level – and therefore having the same number of accidentals in the key signature.
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Bibliography:

Poszowski Antoni Harmonia tonalna, Gdansk 1980
Rasiowa Helena Wstep do matematyki wspolczesnej, Warszawa 1971
Sikorski Kazimierz Harmonia, t. 1-2, Krakow 1972
Targosz Jacek Podstawy harmonii funkcyjnej, Krakow 1993